sdp based approach for graph partitioning and embedding negative type metrics into l 1 subhash khot...
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TRANSCRIPT
SDP Based Approach for Graph
Partitioning and
Embedding Negative Type Metrics into
L1
Subhash Khot (Georgia Tech)
Nisheeth K Vishnoi (IBM Research and Georgia Tech)
CS Perspective
Math Perspective
Parts I amp II
CS Story SparsityS
Sc
Sparsity of a Cut|E(SSc)|----------|S| |Sc|
Sparsest Cut (SC)cut of minimum sparsity b-Balanced Separator (BS)
cut with |S||Sc| ge bn that minimizes |E(SSc)|
b ~ frac12
ApplicationsRelated Measures
Sparsity is often referred to as Graph Conductance
Edge expansion or isoperimetric constant
Applications VLSI Layout
Clustering
Markov Chains
Geometric (Metric) Embeddings
Estimating Sparsity
λ2(G)n le sparsity(G) le 3n radicλ2(G) radic∆
λ2 spectral gap (second eigenvalue of the Laplacian)
Not satisfactory eg n-cycle
Approx Algo For any graph G on n vertices compute a(G) which is within a mult factor f(n)ge1 of sparsity of G
a(G) le sparsity(G) le a(G) f(n)
f(n)=1 is hardWhat about f(n) = log n or even 10
Hard to compute exactly ndash compute ldquoapproximationsrdquo
HistoryAlgorithms
Spectral Graph partitioning Alon-Milman rsquo85 Speilman-Teng rsquo96(eigenvector based)
O(log n) Leighton-Rao rsquo88 (Linear Programming (LP) based)
O(log n) London-Linial-Rabinovitch rsquo94 Aumann-Rabani rsquo94 (connection to metric embeddings)
O(radiclog n) Arora-Rao-Vazirani lsquo04(Semi-Definite Programming (SDP) based)
Hardness NP-Hard Hard to approximate within any constant factor (assuming UGC)
Chawla-Krauthgamer-Kumar-Rabani-Sivakumar lsquo05 Khot-Vishnoi lsquo05
Outline of this Talk Part I
1 Graph PartitioningMotivation amp HistorySDP Approach
2 Embedding Negative Type Metrics into L1 Metric Spaces and EmbeddabilityCut Cone asymp L1
Negative Type Metrics as SDP SolutionsLLRAR Connection
Part II 1 Integrality Gap instance
Hypercube and Cutsbull Kahn-Kalai-Linial (isoperimetry of hypercube)
The GraphThe SDP Solution
2 Conclusion
Outline of this Talk Part I
1 Graph PartitioningMotivation amp HistorySDP Approach
2 Embedding Negative Type Metrics into L1 Metric Spaces and EmbeddabilityCut Cone asymp L1
Negative Type Metrics as SDP SolutionsLLRAR Connection Part II
1 Integrality Gap instance Hypercube and Cuts
bull Kahn-Kalai-Linial (isoperimetry of hypercube)The GraphThe SDP Solution
2 Conclusion
Quadratic Program for BS
i vi -11
Minimize frac14 |vi - vj |2
ij ε E
|vi - vj |2 = n2
iltj
Quadratic ProgramInput G(VE)
Output (SSc)
st
|S||Sc| = n2
which minimizes
|E(SSc)|
Balanced Separator
SDP for Balanced Separator
i vi -11
Minimize frac14 |vi - vj |2
ij ε E
|vi - vj |2 = n2
iltj
Quadratic Program SDP Relaxation
i vi Rn ||vi||=1
Minimize frac14 || vi - vj ||2
ij E
Well-Separatedness
|| vi - vj ||2 = n2
iltj
Why is this a Relaxation SDP Relaxation
G(VE)
i vi Rn ||vi||=1
Minimize frac14 || vi - vj ||2
ij E
Well-Separatedness
|| vi - vj ||2 = n2
iltj
Relaxation
u be a unit vector and (SSc)|S| |Sc| = n2
For i є S vi = u i є Sc vi = - u ||vi-vj||2 = 4 δS(ij) Cost of solution = |E(SSc)| sdp le opt
SDP can be computed in polynomial time
Boils down to the spectral approach Nothing gained()
Quadratic Program for BS hellip
i vi -11
Minimize frac14 |vi - vj |2
ij ε E
|vi - vj |2 = n2
iltj
Triangle Inequality i j k |vi - vj |2
+ |vj - vk |2 |vi - vk|2 (redundant)
Quadratic Program
Input G(VE)
Output (SSc)
st
|S||Sc| = n2
which minimizes
|E(SSc)|
Balanced Separator
SDP for Balanced Separator hellip
i vi -11
Minimize frac14 |vi - vj |2
ij ε E
|vi - vj |2 = n2
iltj
Triangle Inequality i j k |vi - vj |2+|vj - vk |2|vi - vk|2
Quadratic Program SDP Relaxation
i vi Rn ||vi||=1
Minimize frac14 || vi - vj ||2
ij E
Well-Separatedness
|| vi - vj ||2 = n2
iltj
Triangle Inequality i j k ||vi - vj||2 +||vj - vk||2 ||vi - vk||2
Still a relaxation hellip
Geometry of Triangle Ineq
le 90ovi
vk
vj
each step of length 1
t-steps length at-most radict
Rules out the embedding obtained by the spectral method
Integrality Gap Upper Bound
Arora-Rao-Vazirani rsquo04
O(radiclog n) for Sparsest Cut Balanced Separator
sdp within a factor of O(radiclog n) of the opt
Integrality gap max over all graphs on n vertices the ratio of
optsdp (as a function of n)
ARV conjectured that the integrality gap is upper bounded by some constant (independent of n)
Lack of any counterexample
Outline of this Talk Part I
1 Graph PartitioningMotivation amp HistorySDP Approach
2 Embedding Negative Type Metrics into L1 Metric Spaces and EmbeddabilityCut Cone asymp L1
Negative Type Metrics as SDP SolutionsLLRAR Connection
Part II 1 Integrality Gap instance
Hypercube and Cutsbull Kahn-Kalai-Linial (isoperimetry of hypercube)
The GraphThe SDP Solution
2 Conclusion
Math Story Metric Embeddings
bull Metric is a distance function d on [n] x [n] st d(i j) + d(j k) d(i k)
(triangle inequality)
bull Metric d embeds into metric with distortion 1 if there is a map φ st
ij d(i j) (φ(i) φ(j)) d(i j) (distances are preserved upto a factor of )
Negative Type Metrics (squared-L2)
bull d on 12hellipn is of negative type if there are vectors v1 v2 hellip vn
bull Such that d(i j) = || vi - vj ||2 satisfies triangle inequality
bull Same as i j k || vi - vj ||2
+ || vj - vk ||2 || vi - v k||2
bull NEG = class of such metrics bull arise as SDP solutions
bull L1 NEG
Embedding NEG into L1
Conjecture (Explicit by Goemans Linial abt rsquo95)
Every NEG metric embeds into L1 with
O(1) (constant) distortion
Whatrsquos the connection to sparsity
Cuts and L1 Metrics
pS non-negative real for every subset S of [n]
d(ij) = pSδS(ij)
Fact d is isometrically embeddable in L1
Further Every L1 embeddable metric on n-points can be written as non-negative linear sum of cut-metrics on 1hellipn
Cut-Metrics on 12hellipn
δS(i j) = 1 if i j are separated by (SSc) = 0 otherwise
S Sc
Sparsest Cut asymp Optimizing Over L1
[Aumann Rabani 98 Linial London Rabinovich rsquo94]
bull LP-relaxation over all METRICSbull [Bourgain rsquo85] Every n-point metric embeds
into L1 with O(log n) distortionbull O(log n) factor approximation for Sparsity
i~j δS(i j)
---------- iltj δS(i j)
Minimize S V
Minimize d is L1
i~j d(i j)
---------- iiltj d(i j)
=
Metric Embeddings amp Sparsity
bull Optimizing over cuts asymp Optimizing over L1 metrics
bull SDP solution asymp Optimizing over NEG
bull Goemans-LinialARV Conjecture NEG embeds in L1 with O(1) distortion Integrality Gap is O(1)
bull Implies O(1) approx algo for estimating sparsity
Integrality Gap Lower Bound
Sparsest Cut Balanced Separator
log log n integrality gap instance
Khot-Vishnoi rsquo05 Krauthgamer-Rabani lsquo06Devanur-Khot-Saket-Vishnoi lsquo06
bull Disproves the GLARV Conjecture bull Previous best lower bound 116 [Zatloukal rsquo04]
Outline of this Talk Part I
1 Graph PartitioningMotivation amp HistorySDP Approach
2 Embedding Negative Type Metrics into L1 Metric Spaces and EmbeddabilityCut Cone asymp L1
Negative Type Metrics as SDP SolutionsLLRAR Connection Part II
1 Integrality Gap instance Hypercube and Cuts
bull Kahn-Kalai-Linial (isoperimetry of hypercube)The GraphThe SDP Solution
2 Conclusion
Outline of this Talk Part I
1 Graph PartitioningMotivation amp HistorySDP Approach
2 Embedding Negative Type Metrics into L1 Metric Spaces and EmbeddabilityCut Cone asymp L1
Negative Type Metrics as SDP SolutionsLLRAR Connection Part II
1 Integrality Gap instance Hypercube and Cuts
bull Kahn-Kalai-Linial (isoperimetry of hypercube)The GraphThe SDP Solution
2 Conclusion
Recall Integrality Gap Lower Bound
Sparsest Cut Balanced Separator
log log n integrality gap instance
Khot-Vishnoi rsquo05 Krauthgamer-Rabani rsquo06 Devanur-Khot-Saket-Vishnoi lsquo06
bull Disproves the GLARV Conjecture bull Previous best lower bound 116 [Zatloukal rsquo04]
Integrality Gap Lower Bound
Thm Construct graph G(1hellipnE) and unit vector assignment i -gt vi є Rn st
1 G is an ldquoexpanderrdquo every frac14 balanced cut has Ω(|E|(log log n)log n) edges (Kahn-Kalai-Linial)
2 ldquoLowrdquo SDP solution O(|E|log n)
3 Well-Separatedness Σiltj ||vi-vj||2 = n2
4 Triangle inequality d(ij)=||vi-vj||2 is a metric
Integrality gap Ω(log log n)
Starting Point Hypercube()
H=-11k
n = 2k
(111)
(-111)
(11-1)
(1-11)
(1-1-1)
(-1-11)
(-11-1) (-1-1-1)
Hypercube hellip H=-11k Advantages
bull Understand cuts in H tools from Fourier Analysis
bull Vertex is a vector in Rk starting point for SDP solution
But hellip
bull Hypercube has ldquosmallrdquo balanced cuts coordinate cuts have 1k fraction of the edges
Cuts in Hypercube Coordinate of edges = |E(H)|k
Edges across pairs of vertices differing in i-th bit
Cuts in Hypercube hellip
decompose into coordinate cuts
any balanced cut has a coord cut which contributes E(H)k2 edges
Kahn-Kalai-Linial
any balanced cut has a coordinate cut which contributes E(H) (log
k)k2 edges
Increasing Size of Balanced Cuts
consider balanced cuts in which coordinates are indistinguishable
(wrt to their contribution to the cut)
can be achieved by symmetrizing the hypercube
each coordinate contributes equally total E(H) (log k)k edges
eg 4-dim hypercube
(1111)
(-1-1-1-1)
(111-1) (-1111) (1-111) (11-11)
(-1-1-11) (1-1-1-1) (-11-1-1) (-1-11-1)
(11-1-1) (-111-1) (-1-111) (1-1-11)
(1-11-1) (-11-11)
More Formally hellipH=-11k with a rotation group acting on its coordinates
Partitions H into equivalence classes V1hellipVn
Each Vi is a vertex Edges are hypercube edges
G(VE) |E(G)|=|E(H)| k ~ log n
Balanced cuts in G correspond to balanced cuts in H
KKL any balanced cut (in H) has ldquoardquo coordinate cut which contributes E(H) (log k)k2 edges to the cut
Group is transitive ldquoeveryrdquo coordinate cut has the same contribution
Any balanced cut in G has ge E(G) (log k)k = E(G) (log log n)(log n)
Integrality Gap Lower Bound
Thm Construct graph G(1hellipnE) and unit vector assignment i -gt vi є Rn st
1 G is an ldquoexpanderrdquo every frac14 balanced cut has Ω(|E|(log log n)log n) edges (Kahn-Kalai-Linial)
2 ldquoLowrdquo SDP solution O(|E|log n)
3 Well-Separatedness Σiltj ||vi-vj||2 = n2
4 Triangle inequality d(ij)=||vi-vj||2 is a metric
Integrality gap Ω(log log n)
SDP Solution
(1111)
(-1-1-1-1)
(111-1) (-1111) (1-111) (11-11)
(-1-1-11) (1-1-1-1) (-11-1-1) (-1-11-1)
(11-1-1) (-111-1) (-1-111) (1-1-11)
(1-11-1) (-11-11)
Formally SDP SolutionVertex equivalence class x1 x2 hellip xk (rotations)
Vector (1radick) Σj xj
Observations
bull Edge across two nodes differing in one bit
Contribution to sdp ~ 1k sdp le |E(G)|k = |E(G)|(log n)
Triangle Inequality (little bit of work and case analysis)
For most classes x1 hellip xk is ldquonearly orthogonalrdquo
Hidden Gram-Schmidt Orthogonalization Tensoring Well-separatedness
Conclusion
bull (Simple) log log n integrality gap for SCBS
bull Close the gap between log log n and log n
bull [Lee-Naor rsquo06 Cheeger-Kleiner rsquo06] Another counter-example May give (log n)c
Thank you
- Slide 1
- CS Story Sparsity
- Applications
- Estimating Sparsity
- History
- Outline of this Talk
- Slide 7
- Quadratic Program for BS
- SDP for Balanced Separator
- Why is this a Relaxation
- Quadratic Program for BS hellip
- SDP for Balanced Separator hellip
- Geometry of Triangle Ineq
- Integrality Gap Upper Bound
- Slide 15
- Math Story Metric Embeddings
- Negative Type Metrics (squared-L2)
- Embedding NEG into L1
- Cuts and L1 Metrics
- Sparsest Cut asymp Optimizing Over L1
- Metric Embeddings amp Sparsity
- Integrality Gap Lower Bound
- Slide 23
- Slide 24
- Recall Integrality Gap Lower Bound
- Slide 26
- Starting Point Hypercube()
- Hypercube hellip
- Cuts in Hypercube Coordinate
- Cuts in Hypercube hellip
- Kahn-Kalai-Linial
- Increasing Size of Balanced Cuts
- eg 4-dim hypercube
- More Formally hellip
- Slide 35
- SDP Solution
- Formally SDP Solution
- Conclusion
-
CS Story SparsityS
Sc
Sparsity of a Cut|E(SSc)|----------|S| |Sc|
Sparsest Cut (SC)cut of minimum sparsity b-Balanced Separator (BS)
cut with |S||Sc| ge bn that minimizes |E(SSc)|
b ~ frac12
ApplicationsRelated Measures
Sparsity is often referred to as Graph Conductance
Edge expansion or isoperimetric constant
Applications VLSI Layout
Clustering
Markov Chains
Geometric (Metric) Embeddings
Estimating Sparsity
λ2(G)n le sparsity(G) le 3n radicλ2(G) radic∆
λ2 spectral gap (second eigenvalue of the Laplacian)
Not satisfactory eg n-cycle
Approx Algo For any graph G on n vertices compute a(G) which is within a mult factor f(n)ge1 of sparsity of G
a(G) le sparsity(G) le a(G) f(n)
f(n)=1 is hardWhat about f(n) = log n or even 10
Hard to compute exactly ndash compute ldquoapproximationsrdquo
HistoryAlgorithms
Spectral Graph partitioning Alon-Milman rsquo85 Speilman-Teng rsquo96(eigenvector based)
O(log n) Leighton-Rao rsquo88 (Linear Programming (LP) based)
O(log n) London-Linial-Rabinovitch rsquo94 Aumann-Rabani rsquo94 (connection to metric embeddings)
O(radiclog n) Arora-Rao-Vazirani lsquo04(Semi-Definite Programming (SDP) based)
Hardness NP-Hard Hard to approximate within any constant factor (assuming UGC)
Chawla-Krauthgamer-Kumar-Rabani-Sivakumar lsquo05 Khot-Vishnoi lsquo05
Outline of this Talk Part I
1 Graph PartitioningMotivation amp HistorySDP Approach
2 Embedding Negative Type Metrics into L1 Metric Spaces and EmbeddabilityCut Cone asymp L1
Negative Type Metrics as SDP SolutionsLLRAR Connection
Part II 1 Integrality Gap instance
Hypercube and Cutsbull Kahn-Kalai-Linial (isoperimetry of hypercube)
The GraphThe SDP Solution
2 Conclusion
Outline of this Talk Part I
1 Graph PartitioningMotivation amp HistorySDP Approach
2 Embedding Negative Type Metrics into L1 Metric Spaces and EmbeddabilityCut Cone asymp L1
Negative Type Metrics as SDP SolutionsLLRAR Connection Part II
1 Integrality Gap instance Hypercube and Cuts
bull Kahn-Kalai-Linial (isoperimetry of hypercube)The GraphThe SDP Solution
2 Conclusion
Quadratic Program for BS
i vi -11
Minimize frac14 |vi - vj |2
ij ε E
|vi - vj |2 = n2
iltj
Quadratic ProgramInput G(VE)
Output (SSc)
st
|S||Sc| = n2
which minimizes
|E(SSc)|
Balanced Separator
SDP for Balanced Separator
i vi -11
Minimize frac14 |vi - vj |2
ij ε E
|vi - vj |2 = n2
iltj
Quadratic Program SDP Relaxation
i vi Rn ||vi||=1
Minimize frac14 || vi - vj ||2
ij E
Well-Separatedness
|| vi - vj ||2 = n2
iltj
Why is this a Relaxation SDP Relaxation
G(VE)
i vi Rn ||vi||=1
Minimize frac14 || vi - vj ||2
ij E
Well-Separatedness
|| vi - vj ||2 = n2
iltj
Relaxation
u be a unit vector and (SSc)|S| |Sc| = n2
For i є S vi = u i є Sc vi = - u ||vi-vj||2 = 4 δS(ij) Cost of solution = |E(SSc)| sdp le opt
SDP can be computed in polynomial time
Boils down to the spectral approach Nothing gained()
Quadratic Program for BS hellip
i vi -11
Minimize frac14 |vi - vj |2
ij ε E
|vi - vj |2 = n2
iltj
Triangle Inequality i j k |vi - vj |2
+ |vj - vk |2 |vi - vk|2 (redundant)
Quadratic Program
Input G(VE)
Output (SSc)
st
|S||Sc| = n2
which minimizes
|E(SSc)|
Balanced Separator
SDP for Balanced Separator hellip
i vi -11
Minimize frac14 |vi - vj |2
ij ε E
|vi - vj |2 = n2
iltj
Triangle Inequality i j k |vi - vj |2+|vj - vk |2|vi - vk|2
Quadratic Program SDP Relaxation
i vi Rn ||vi||=1
Minimize frac14 || vi - vj ||2
ij E
Well-Separatedness
|| vi - vj ||2 = n2
iltj
Triangle Inequality i j k ||vi - vj||2 +||vj - vk||2 ||vi - vk||2
Still a relaxation hellip
Geometry of Triangle Ineq
le 90ovi
vk
vj
each step of length 1
t-steps length at-most radict
Rules out the embedding obtained by the spectral method
Integrality Gap Upper Bound
Arora-Rao-Vazirani rsquo04
O(radiclog n) for Sparsest Cut Balanced Separator
sdp within a factor of O(radiclog n) of the opt
Integrality gap max over all graphs on n vertices the ratio of
optsdp (as a function of n)
ARV conjectured that the integrality gap is upper bounded by some constant (independent of n)
Lack of any counterexample
Outline of this Talk Part I
1 Graph PartitioningMotivation amp HistorySDP Approach
2 Embedding Negative Type Metrics into L1 Metric Spaces and EmbeddabilityCut Cone asymp L1
Negative Type Metrics as SDP SolutionsLLRAR Connection
Part II 1 Integrality Gap instance
Hypercube and Cutsbull Kahn-Kalai-Linial (isoperimetry of hypercube)
The GraphThe SDP Solution
2 Conclusion
Math Story Metric Embeddings
bull Metric is a distance function d on [n] x [n] st d(i j) + d(j k) d(i k)
(triangle inequality)
bull Metric d embeds into metric with distortion 1 if there is a map φ st
ij d(i j) (φ(i) φ(j)) d(i j) (distances are preserved upto a factor of )
Negative Type Metrics (squared-L2)
bull d on 12hellipn is of negative type if there are vectors v1 v2 hellip vn
bull Such that d(i j) = || vi - vj ||2 satisfies triangle inequality
bull Same as i j k || vi - vj ||2
+ || vj - vk ||2 || vi - v k||2
bull NEG = class of such metrics bull arise as SDP solutions
bull L1 NEG
Embedding NEG into L1
Conjecture (Explicit by Goemans Linial abt rsquo95)
Every NEG metric embeds into L1 with
O(1) (constant) distortion
Whatrsquos the connection to sparsity
Cuts and L1 Metrics
pS non-negative real for every subset S of [n]
d(ij) = pSδS(ij)
Fact d is isometrically embeddable in L1
Further Every L1 embeddable metric on n-points can be written as non-negative linear sum of cut-metrics on 1hellipn
Cut-Metrics on 12hellipn
δS(i j) = 1 if i j are separated by (SSc) = 0 otherwise
S Sc
Sparsest Cut asymp Optimizing Over L1
[Aumann Rabani 98 Linial London Rabinovich rsquo94]
bull LP-relaxation over all METRICSbull [Bourgain rsquo85] Every n-point metric embeds
into L1 with O(log n) distortionbull O(log n) factor approximation for Sparsity
i~j δS(i j)
---------- iltj δS(i j)
Minimize S V
Minimize d is L1
i~j d(i j)
---------- iiltj d(i j)
=
Metric Embeddings amp Sparsity
bull Optimizing over cuts asymp Optimizing over L1 metrics
bull SDP solution asymp Optimizing over NEG
bull Goemans-LinialARV Conjecture NEG embeds in L1 with O(1) distortion Integrality Gap is O(1)
bull Implies O(1) approx algo for estimating sparsity
Integrality Gap Lower Bound
Sparsest Cut Balanced Separator
log log n integrality gap instance
Khot-Vishnoi rsquo05 Krauthgamer-Rabani lsquo06Devanur-Khot-Saket-Vishnoi lsquo06
bull Disproves the GLARV Conjecture bull Previous best lower bound 116 [Zatloukal rsquo04]
Outline of this Talk Part I
1 Graph PartitioningMotivation amp HistorySDP Approach
2 Embedding Negative Type Metrics into L1 Metric Spaces and EmbeddabilityCut Cone asymp L1
Negative Type Metrics as SDP SolutionsLLRAR Connection Part II
1 Integrality Gap instance Hypercube and Cuts
bull Kahn-Kalai-Linial (isoperimetry of hypercube)The GraphThe SDP Solution
2 Conclusion
Outline of this Talk Part I
1 Graph PartitioningMotivation amp HistorySDP Approach
2 Embedding Negative Type Metrics into L1 Metric Spaces and EmbeddabilityCut Cone asymp L1
Negative Type Metrics as SDP SolutionsLLRAR Connection Part II
1 Integrality Gap instance Hypercube and Cuts
bull Kahn-Kalai-Linial (isoperimetry of hypercube)The GraphThe SDP Solution
2 Conclusion
Recall Integrality Gap Lower Bound
Sparsest Cut Balanced Separator
log log n integrality gap instance
Khot-Vishnoi rsquo05 Krauthgamer-Rabani rsquo06 Devanur-Khot-Saket-Vishnoi lsquo06
bull Disproves the GLARV Conjecture bull Previous best lower bound 116 [Zatloukal rsquo04]
Integrality Gap Lower Bound
Thm Construct graph G(1hellipnE) and unit vector assignment i -gt vi є Rn st
1 G is an ldquoexpanderrdquo every frac14 balanced cut has Ω(|E|(log log n)log n) edges (Kahn-Kalai-Linial)
2 ldquoLowrdquo SDP solution O(|E|log n)
3 Well-Separatedness Σiltj ||vi-vj||2 = n2
4 Triangle inequality d(ij)=||vi-vj||2 is a metric
Integrality gap Ω(log log n)
Starting Point Hypercube()
H=-11k
n = 2k
(111)
(-111)
(11-1)
(1-11)
(1-1-1)
(-1-11)
(-11-1) (-1-1-1)
Hypercube hellip H=-11k Advantages
bull Understand cuts in H tools from Fourier Analysis
bull Vertex is a vector in Rk starting point for SDP solution
But hellip
bull Hypercube has ldquosmallrdquo balanced cuts coordinate cuts have 1k fraction of the edges
Cuts in Hypercube Coordinate of edges = |E(H)|k
Edges across pairs of vertices differing in i-th bit
Cuts in Hypercube hellip
decompose into coordinate cuts
any balanced cut has a coord cut which contributes E(H)k2 edges
Kahn-Kalai-Linial
any balanced cut has a coordinate cut which contributes E(H) (log
k)k2 edges
Increasing Size of Balanced Cuts
consider balanced cuts in which coordinates are indistinguishable
(wrt to their contribution to the cut)
can be achieved by symmetrizing the hypercube
each coordinate contributes equally total E(H) (log k)k edges
eg 4-dim hypercube
(1111)
(-1-1-1-1)
(111-1) (-1111) (1-111) (11-11)
(-1-1-11) (1-1-1-1) (-11-1-1) (-1-11-1)
(11-1-1) (-111-1) (-1-111) (1-1-11)
(1-11-1) (-11-11)
More Formally hellipH=-11k with a rotation group acting on its coordinates
Partitions H into equivalence classes V1hellipVn
Each Vi is a vertex Edges are hypercube edges
G(VE) |E(G)|=|E(H)| k ~ log n
Balanced cuts in G correspond to balanced cuts in H
KKL any balanced cut (in H) has ldquoardquo coordinate cut which contributes E(H) (log k)k2 edges to the cut
Group is transitive ldquoeveryrdquo coordinate cut has the same contribution
Any balanced cut in G has ge E(G) (log k)k = E(G) (log log n)(log n)
Integrality Gap Lower Bound
Thm Construct graph G(1hellipnE) and unit vector assignment i -gt vi є Rn st
1 G is an ldquoexpanderrdquo every frac14 balanced cut has Ω(|E|(log log n)log n) edges (Kahn-Kalai-Linial)
2 ldquoLowrdquo SDP solution O(|E|log n)
3 Well-Separatedness Σiltj ||vi-vj||2 = n2
4 Triangle inequality d(ij)=||vi-vj||2 is a metric
Integrality gap Ω(log log n)
SDP Solution
(1111)
(-1-1-1-1)
(111-1) (-1111) (1-111) (11-11)
(-1-1-11) (1-1-1-1) (-11-1-1) (-1-11-1)
(11-1-1) (-111-1) (-1-111) (1-1-11)
(1-11-1) (-11-11)
Formally SDP SolutionVertex equivalence class x1 x2 hellip xk (rotations)
Vector (1radick) Σj xj
Observations
bull Edge across two nodes differing in one bit
Contribution to sdp ~ 1k sdp le |E(G)|k = |E(G)|(log n)
Triangle Inequality (little bit of work and case analysis)
For most classes x1 hellip xk is ldquonearly orthogonalrdquo
Hidden Gram-Schmidt Orthogonalization Tensoring Well-separatedness
Conclusion
bull (Simple) log log n integrality gap for SCBS
bull Close the gap between log log n and log n
bull [Lee-Naor rsquo06 Cheeger-Kleiner rsquo06] Another counter-example May give (log n)c
Thank you
- Slide 1
- CS Story Sparsity
- Applications
- Estimating Sparsity
- History
- Outline of this Talk
- Slide 7
- Quadratic Program for BS
- SDP for Balanced Separator
- Why is this a Relaxation
- Quadratic Program for BS hellip
- SDP for Balanced Separator hellip
- Geometry of Triangle Ineq
- Integrality Gap Upper Bound
- Slide 15
- Math Story Metric Embeddings
- Negative Type Metrics (squared-L2)
- Embedding NEG into L1
- Cuts and L1 Metrics
- Sparsest Cut asymp Optimizing Over L1
- Metric Embeddings amp Sparsity
- Integrality Gap Lower Bound
- Slide 23
- Slide 24
- Recall Integrality Gap Lower Bound
- Slide 26
- Starting Point Hypercube()
- Hypercube hellip
- Cuts in Hypercube Coordinate
- Cuts in Hypercube hellip
- Kahn-Kalai-Linial
- Increasing Size of Balanced Cuts
- eg 4-dim hypercube
- More Formally hellip
- Slide 35
- SDP Solution
- Formally SDP Solution
- Conclusion
-
ApplicationsRelated Measures
Sparsity is often referred to as Graph Conductance
Edge expansion or isoperimetric constant
Applications VLSI Layout
Clustering
Markov Chains
Geometric (Metric) Embeddings
Estimating Sparsity
λ2(G)n le sparsity(G) le 3n radicλ2(G) radic∆
λ2 spectral gap (second eigenvalue of the Laplacian)
Not satisfactory eg n-cycle
Approx Algo For any graph G on n vertices compute a(G) which is within a mult factor f(n)ge1 of sparsity of G
a(G) le sparsity(G) le a(G) f(n)
f(n)=1 is hardWhat about f(n) = log n or even 10
Hard to compute exactly ndash compute ldquoapproximationsrdquo
HistoryAlgorithms
Spectral Graph partitioning Alon-Milman rsquo85 Speilman-Teng rsquo96(eigenvector based)
O(log n) Leighton-Rao rsquo88 (Linear Programming (LP) based)
O(log n) London-Linial-Rabinovitch rsquo94 Aumann-Rabani rsquo94 (connection to metric embeddings)
O(radiclog n) Arora-Rao-Vazirani lsquo04(Semi-Definite Programming (SDP) based)
Hardness NP-Hard Hard to approximate within any constant factor (assuming UGC)
Chawla-Krauthgamer-Kumar-Rabani-Sivakumar lsquo05 Khot-Vishnoi lsquo05
Outline of this Talk Part I
1 Graph PartitioningMotivation amp HistorySDP Approach
2 Embedding Negative Type Metrics into L1 Metric Spaces and EmbeddabilityCut Cone asymp L1
Negative Type Metrics as SDP SolutionsLLRAR Connection
Part II 1 Integrality Gap instance
Hypercube and Cutsbull Kahn-Kalai-Linial (isoperimetry of hypercube)
The GraphThe SDP Solution
2 Conclusion
Outline of this Talk Part I
1 Graph PartitioningMotivation amp HistorySDP Approach
2 Embedding Negative Type Metrics into L1 Metric Spaces and EmbeddabilityCut Cone asymp L1
Negative Type Metrics as SDP SolutionsLLRAR Connection Part II
1 Integrality Gap instance Hypercube and Cuts
bull Kahn-Kalai-Linial (isoperimetry of hypercube)The GraphThe SDP Solution
2 Conclusion
Quadratic Program for BS
i vi -11
Minimize frac14 |vi - vj |2
ij ε E
|vi - vj |2 = n2
iltj
Quadratic ProgramInput G(VE)
Output (SSc)
st
|S||Sc| = n2
which minimizes
|E(SSc)|
Balanced Separator
SDP for Balanced Separator
i vi -11
Minimize frac14 |vi - vj |2
ij ε E
|vi - vj |2 = n2
iltj
Quadratic Program SDP Relaxation
i vi Rn ||vi||=1
Minimize frac14 || vi - vj ||2
ij E
Well-Separatedness
|| vi - vj ||2 = n2
iltj
Why is this a Relaxation SDP Relaxation
G(VE)
i vi Rn ||vi||=1
Minimize frac14 || vi - vj ||2
ij E
Well-Separatedness
|| vi - vj ||2 = n2
iltj
Relaxation
u be a unit vector and (SSc)|S| |Sc| = n2
For i є S vi = u i є Sc vi = - u ||vi-vj||2 = 4 δS(ij) Cost of solution = |E(SSc)| sdp le opt
SDP can be computed in polynomial time
Boils down to the spectral approach Nothing gained()
Quadratic Program for BS hellip
i vi -11
Minimize frac14 |vi - vj |2
ij ε E
|vi - vj |2 = n2
iltj
Triangle Inequality i j k |vi - vj |2
+ |vj - vk |2 |vi - vk|2 (redundant)
Quadratic Program
Input G(VE)
Output (SSc)
st
|S||Sc| = n2
which minimizes
|E(SSc)|
Balanced Separator
SDP for Balanced Separator hellip
i vi -11
Minimize frac14 |vi - vj |2
ij ε E
|vi - vj |2 = n2
iltj
Triangle Inequality i j k |vi - vj |2+|vj - vk |2|vi - vk|2
Quadratic Program SDP Relaxation
i vi Rn ||vi||=1
Minimize frac14 || vi - vj ||2
ij E
Well-Separatedness
|| vi - vj ||2 = n2
iltj
Triangle Inequality i j k ||vi - vj||2 +||vj - vk||2 ||vi - vk||2
Still a relaxation hellip
Geometry of Triangle Ineq
le 90ovi
vk
vj
each step of length 1
t-steps length at-most radict
Rules out the embedding obtained by the spectral method
Integrality Gap Upper Bound
Arora-Rao-Vazirani rsquo04
O(radiclog n) for Sparsest Cut Balanced Separator
sdp within a factor of O(radiclog n) of the opt
Integrality gap max over all graphs on n vertices the ratio of
optsdp (as a function of n)
ARV conjectured that the integrality gap is upper bounded by some constant (independent of n)
Lack of any counterexample
Outline of this Talk Part I
1 Graph PartitioningMotivation amp HistorySDP Approach
2 Embedding Negative Type Metrics into L1 Metric Spaces and EmbeddabilityCut Cone asymp L1
Negative Type Metrics as SDP SolutionsLLRAR Connection
Part II 1 Integrality Gap instance
Hypercube and Cutsbull Kahn-Kalai-Linial (isoperimetry of hypercube)
The GraphThe SDP Solution
2 Conclusion
Math Story Metric Embeddings
bull Metric is a distance function d on [n] x [n] st d(i j) + d(j k) d(i k)
(triangle inequality)
bull Metric d embeds into metric with distortion 1 if there is a map φ st
ij d(i j) (φ(i) φ(j)) d(i j) (distances are preserved upto a factor of )
Negative Type Metrics (squared-L2)
bull d on 12hellipn is of negative type if there are vectors v1 v2 hellip vn
bull Such that d(i j) = || vi - vj ||2 satisfies triangle inequality
bull Same as i j k || vi - vj ||2
+ || vj - vk ||2 || vi - v k||2
bull NEG = class of such metrics bull arise as SDP solutions
bull L1 NEG
Embedding NEG into L1
Conjecture (Explicit by Goemans Linial abt rsquo95)
Every NEG metric embeds into L1 with
O(1) (constant) distortion
Whatrsquos the connection to sparsity
Cuts and L1 Metrics
pS non-negative real for every subset S of [n]
d(ij) = pSδS(ij)
Fact d is isometrically embeddable in L1
Further Every L1 embeddable metric on n-points can be written as non-negative linear sum of cut-metrics on 1hellipn
Cut-Metrics on 12hellipn
δS(i j) = 1 if i j are separated by (SSc) = 0 otherwise
S Sc
Sparsest Cut asymp Optimizing Over L1
[Aumann Rabani 98 Linial London Rabinovich rsquo94]
bull LP-relaxation over all METRICSbull [Bourgain rsquo85] Every n-point metric embeds
into L1 with O(log n) distortionbull O(log n) factor approximation for Sparsity
i~j δS(i j)
---------- iltj δS(i j)
Minimize S V
Minimize d is L1
i~j d(i j)
---------- iiltj d(i j)
=
Metric Embeddings amp Sparsity
bull Optimizing over cuts asymp Optimizing over L1 metrics
bull SDP solution asymp Optimizing over NEG
bull Goemans-LinialARV Conjecture NEG embeds in L1 with O(1) distortion Integrality Gap is O(1)
bull Implies O(1) approx algo for estimating sparsity
Integrality Gap Lower Bound
Sparsest Cut Balanced Separator
log log n integrality gap instance
Khot-Vishnoi rsquo05 Krauthgamer-Rabani lsquo06Devanur-Khot-Saket-Vishnoi lsquo06
bull Disproves the GLARV Conjecture bull Previous best lower bound 116 [Zatloukal rsquo04]
Outline of this Talk Part I
1 Graph PartitioningMotivation amp HistorySDP Approach
2 Embedding Negative Type Metrics into L1 Metric Spaces and EmbeddabilityCut Cone asymp L1
Negative Type Metrics as SDP SolutionsLLRAR Connection Part II
1 Integrality Gap instance Hypercube and Cuts
bull Kahn-Kalai-Linial (isoperimetry of hypercube)The GraphThe SDP Solution
2 Conclusion
Outline of this Talk Part I
1 Graph PartitioningMotivation amp HistorySDP Approach
2 Embedding Negative Type Metrics into L1 Metric Spaces and EmbeddabilityCut Cone asymp L1
Negative Type Metrics as SDP SolutionsLLRAR Connection Part II
1 Integrality Gap instance Hypercube and Cuts
bull Kahn-Kalai-Linial (isoperimetry of hypercube)The GraphThe SDP Solution
2 Conclusion
Recall Integrality Gap Lower Bound
Sparsest Cut Balanced Separator
log log n integrality gap instance
Khot-Vishnoi rsquo05 Krauthgamer-Rabani rsquo06 Devanur-Khot-Saket-Vishnoi lsquo06
bull Disproves the GLARV Conjecture bull Previous best lower bound 116 [Zatloukal rsquo04]
Integrality Gap Lower Bound
Thm Construct graph G(1hellipnE) and unit vector assignment i -gt vi є Rn st
1 G is an ldquoexpanderrdquo every frac14 balanced cut has Ω(|E|(log log n)log n) edges (Kahn-Kalai-Linial)
2 ldquoLowrdquo SDP solution O(|E|log n)
3 Well-Separatedness Σiltj ||vi-vj||2 = n2
4 Triangle inequality d(ij)=||vi-vj||2 is a metric
Integrality gap Ω(log log n)
Starting Point Hypercube()
H=-11k
n = 2k
(111)
(-111)
(11-1)
(1-11)
(1-1-1)
(-1-11)
(-11-1) (-1-1-1)
Hypercube hellip H=-11k Advantages
bull Understand cuts in H tools from Fourier Analysis
bull Vertex is a vector in Rk starting point for SDP solution
But hellip
bull Hypercube has ldquosmallrdquo balanced cuts coordinate cuts have 1k fraction of the edges
Cuts in Hypercube Coordinate of edges = |E(H)|k
Edges across pairs of vertices differing in i-th bit
Cuts in Hypercube hellip
decompose into coordinate cuts
any balanced cut has a coord cut which contributes E(H)k2 edges
Kahn-Kalai-Linial
any balanced cut has a coordinate cut which contributes E(H) (log
k)k2 edges
Increasing Size of Balanced Cuts
consider balanced cuts in which coordinates are indistinguishable
(wrt to their contribution to the cut)
can be achieved by symmetrizing the hypercube
each coordinate contributes equally total E(H) (log k)k edges
eg 4-dim hypercube
(1111)
(-1-1-1-1)
(111-1) (-1111) (1-111) (11-11)
(-1-1-11) (1-1-1-1) (-11-1-1) (-1-11-1)
(11-1-1) (-111-1) (-1-111) (1-1-11)
(1-11-1) (-11-11)
More Formally hellipH=-11k with a rotation group acting on its coordinates
Partitions H into equivalence classes V1hellipVn
Each Vi is a vertex Edges are hypercube edges
G(VE) |E(G)|=|E(H)| k ~ log n
Balanced cuts in G correspond to balanced cuts in H
KKL any balanced cut (in H) has ldquoardquo coordinate cut which contributes E(H) (log k)k2 edges to the cut
Group is transitive ldquoeveryrdquo coordinate cut has the same contribution
Any balanced cut in G has ge E(G) (log k)k = E(G) (log log n)(log n)
Integrality Gap Lower Bound
Thm Construct graph G(1hellipnE) and unit vector assignment i -gt vi є Rn st
1 G is an ldquoexpanderrdquo every frac14 balanced cut has Ω(|E|(log log n)log n) edges (Kahn-Kalai-Linial)
2 ldquoLowrdquo SDP solution O(|E|log n)
3 Well-Separatedness Σiltj ||vi-vj||2 = n2
4 Triangle inequality d(ij)=||vi-vj||2 is a metric
Integrality gap Ω(log log n)
SDP Solution
(1111)
(-1-1-1-1)
(111-1) (-1111) (1-111) (11-11)
(-1-1-11) (1-1-1-1) (-11-1-1) (-1-11-1)
(11-1-1) (-111-1) (-1-111) (1-1-11)
(1-11-1) (-11-11)
Formally SDP SolutionVertex equivalence class x1 x2 hellip xk (rotations)
Vector (1radick) Σj xj
Observations
bull Edge across two nodes differing in one bit
Contribution to sdp ~ 1k sdp le |E(G)|k = |E(G)|(log n)
Triangle Inequality (little bit of work and case analysis)
For most classes x1 hellip xk is ldquonearly orthogonalrdquo
Hidden Gram-Schmidt Orthogonalization Tensoring Well-separatedness
Conclusion
bull (Simple) log log n integrality gap for SCBS
bull Close the gap between log log n and log n
bull [Lee-Naor rsquo06 Cheeger-Kleiner rsquo06] Another counter-example May give (log n)c
Thank you
- Slide 1
- CS Story Sparsity
- Applications
- Estimating Sparsity
- History
- Outline of this Talk
- Slide 7
- Quadratic Program for BS
- SDP for Balanced Separator
- Why is this a Relaxation
- Quadratic Program for BS hellip
- SDP for Balanced Separator hellip
- Geometry of Triangle Ineq
- Integrality Gap Upper Bound
- Slide 15
- Math Story Metric Embeddings
- Negative Type Metrics (squared-L2)
- Embedding NEG into L1
- Cuts and L1 Metrics
- Sparsest Cut asymp Optimizing Over L1
- Metric Embeddings amp Sparsity
- Integrality Gap Lower Bound
- Slide 23
- Slide 24
- Recall Integrality Gap Lower Bound
- Slide 26
- Starting Point Hypercube()
- Hypercube hellip
- Cuts in Hypercube Coordinate
- Cuts in Hypercube hellip
- Kahn-Kalai-Linial
- Increasing Size of Balanced Cuts
- eg 4-dim hypercube
- More Formally hellip
- Slide 35
- SDP Solution
- Formally SDP Solution
- Conclusion
-
Estimating Sparsity
λ2(G)n le sparsity(G) le 3n radicλ2(G) radic∆
λ2 spectral gap (second eigenvalue of the Laplacian)
Not satisfactory eg n-cycle
Approx Algo For any graph G on n vertices compute a(G) which is within a mult factor f(n)ge1 of sparsity of G
a(G) le sparsity(G) le a(G) f(n)
f(n)=1 is hardWhat about f(n) = log n or even 10
Hard to compute exactly ndash compute ldquoapproximationsrdquo
HistoryAlgorithms
Spectral Graph partitioning Alon-Milman rsquo85 Speilman-Teng rsquo96(eigenvector based)
O(log n) Leighton-Rao rsquo88 (Linear Programming (LP) based)
O(log n) London-Linial-Rabinovitch rsquo94 Aumann-Rabani rsquo94 (connection to metric embeddings)
O(radiclog n) Arora-Rao-Vazirani lsquo04(Semi-Definite Programming (SDP) based)
Hardness NP-Hard Hard to approximate within any constant factor (assuming UGC)
Chawla-Krauthgamer-Kumar-Rabani-Sivakumar lsquo05 Khot-Vishnoi lsquo05
Outline of this Talk Part I
1 Graph PartitioningMotivation amp HistorySDP Approach
2 Embedding Negative Type Metrics into L1 Metric Spaces and EmbeddabilityCut Cone asymp L1
Negative Type Metrics as SDP SolutionsLLRAR Connection
Part II 1 Integrality Gap instance
Hypercube and Cutsbull Kahn-Kalai-Linial (isoperimetry of hypercube)
The GraphThe SDP Solution
2 Conclusion
Outline of this Talk Part I
1 Graph PartitioningMotivation amp HistorySDP Approach
2 Embedding Negative Type Metrics into L1 Metric Spaces and EmbeddabilityCut Cone asymp L1
Negative Type Metrics as SDP SolutionsLLRAR Connection Part II
1 Integrality Gap instance Hypercube and Cuts
bull Kahn-Kalai-Linial (isoperimetry of hypercube)The GraphThe SDP Solution
2 Conclusion
Quadratic Program for BS
i vi -11
Minimize frac14 |vi - vj |2
ij ε E
|vi - vj |2 = n2
iltj
Quadratic ProgramInput G(VE)
Output (SSc)
st
|S||Sc| = n2
which minimizes
|E(SSc)|
Balanced Separator
SDP for Balanced Separator
i vi -11
Minimize frac14 |vi - vj |2
ij ε E
|vi - vj |2 = n2
iltj
Quadratic Program SDP Relaxation
i vi Rn ||vi||=1
Minimize frac14 || vi - vj ||2
ij E
Well-Separatedness
|| vi - vj ||2 = n2
iltj
Why is this a Relaxation SDP Relaxation
G(VE)
i vi Rn ||vi||=1
Minimize frac14 || vi - vj ||2
ij E
Well-Separatedness
|| vi - vj ||2 = n2
iltj
Relaxation
u be a unit vector and (SSc)|S| |Sc| = n2
For i є S vi = u i є Sc vi = - u ||vi-vj||2 = 4 δS(ij) Cost of solution = |E(SSc)| sdp le opt
SDP can be computed in polynomial time
Boils down to the spectral approach Nothing gained()
Quadratic Program for BS hellip
i vi -11
Minimize frac14 |vi - vj |2
ij ε E
|vi - vj |2 = n2
iltj
Triangle Inequality i j k |vi - vj |2
+ |vj - vk |2 |vi - vk|2 (redundant)
Quadratic Program
Input G(VE)
Output (SSc)
st
|S||Sc| = n2
which minimizes
|E(SSc)|
Balanced Separator
SDP for Balanced Separator hellip
i vi -11
Minimize frac14 |vi - vj |2
ij ε E
|vi - vj |2 = n2
iltj
Triangle Inequality i j k |vi - vj |2+|vj - vk |2|vi - vk|2
Quadratic Program SDP Relaxation
i vi Rn ||vi||=1
Minimize frac14 || vi - vj ||2
ij E
Well-Separatedness
|| vi - vj ||2 = n2
iltj
Triangle Inequality i j k ||vi - vj||2 +||vj - vk||2 ||vi - vk||2
Still a relaxation hellip
Geometry of Triangle Ineq
le 90ovi
vk
vj
each step of length 1
t-steps length at-most radict
Rules out the embedding obtained by the spectral method
Integrality Gap Upper Bound
Arora-Rao-Vazirani rsquo04
O(radiclog n) for Sparsest Cut Balanced Separator
sdp within a factor of O(radiclog n) of the opt
Integrality gap max over all graphs on n vertices the ratio of
optsdp (as a function of n)
ARV conjectured that the integrality gap is upper bounded by some constant (independent of n)
Lack of any counterexample
Outline of this Talk Part I
1 Graph PartitioningMotivation amp HistorySDP Approach
2 Embedding Negative Type Metrics into L1 Metric Spaces and EmbeddabilityCut Cone asymp L1
Negative Type Metrics as SDP SolutionsLLRAR Connection
Part II 1 Integrality Gap instance
Hypercube and Cutsbull Kahn-Kalai-Linial (isoperimetry of hypercube)
The GraphThe SDP Solution
2 Conclusion
Math Story Metric Embeddings
bull Metric is a distance function d on [n] x [n] st d(i j) + d(j k) d(i k)
(triangle inequality)
bull Metric d embeds into metric with distortion 1 if there is a map φ st
ij d(i j) (φ(i) φ(j)) d(i j) (distances are preserved upto a factor of )
Negative Type Metrics (squared-L2)
bull d on 12hellipn is of negative type if there are vectors v1 v2 hellip vn
bull Such that d(i j) = || vi - vj ||2 satisfies triangle inequality
bull Same as i j k || vi - vj ||2
+ || vj - vk ||2 || vi - v k||2
bull NEG = class of such metrics bull arise as SDP solutions
bull L1 NEG
Embedding NEG into L1
Conjecture (Explicit by Goemans Linial abt rsquo95)
Every NEG metric embeds into L1 with
O(1) (constant) distortion
Whatrsquos the connection to sparsity
Cuts and L1 Metrics
pS non-negative real for every subset S of [n]
d(ij) = pSδS(ij)
Fact d is isometrically embeddable in L1
Further Every L1 embeddable metric on n-points can be written as non-negative linear sum of cut-metrics on 1hellipn
Cut-Metrics on 12hellipn
δS(i j) = 1 if i j are separated by (SSc) = 0 otherwise
S Sc
Sparsest Cut asymp Optimizing Over L1
[Aumann Rabani 98 Linial London Rabinovich rsquo94]
bull LP-relaxation over all METRICSbull [Bourgain rsquo85] Every n-point metric embeds
into L1 with O(log n) distortionbull O(log n) factor approximation for Sparsity
i~j δS(i j)
---------- iltj δS(i j)
Minimize S V
Minimize d is L1
i~j d(i j)
---------- iiltj d(i j)
=
Metric Embeddings amp Sparsity
bull Optimizing over cuts asymp Optimizing over L1 metrics
bull SDP solution asymp Optimizing over NEG
bull Goemans-LinialARV Conjecture NEG embeds in L1 with O(1) distortion Integrality Gap is O(1)
bull Implies O(1) approx algo for estimating sparsity
Integrality Gap Lower Bound
Sparsest Cut Balanced Separator
log log n integrality gap instance
Khot-Vishnoi rsquo05 Krauthgamer-Rabani lsquo06Devanur-Khot-Saket-Vishnoi lsquo06
bull Disproves the GLARV Conjecture bull Previous best lower bound 116 [Zatloukal rsquo04]
Outline of this Talk Part I
1 Graph PartitioningMotivation amp HistorySDP Approach
2 Embedding Negative Type Metrics into L1 Metric Spaces and EmbeddabilityCut Cone asymp L1
Negative Type Metrics as SDP SolutionsLLRAR Connection Part II
1 Integrality Gap instance Hypercube and Cuts
bull Kahn-Kalai-Linial (isoperimetry of hypercube)The GraphThe SDP Solution
2 Conclusion
Outline of this Talk Part I
1 Graph PartitioningMotivation amp HistorySDP Approach
2 Embedding Negative Type Metrics into L1 Metric Spaces and EmbeddabilityCut Cone asymp L1
Negative Type Metrics as SDP SolutionsLLRAR Connection Part II
1 Integrality Gap instance Hypercube and Cuts
bull Kahn-Kalai-Linial (isoperimetry of hypercube)The GraphThe SDP Solution
2 Conclusion
Recall Integrality Gap Lower Bound
Sparsest Cut Balanced Separator
log log n integrality gap instance
Khot-Vishnoi rsquo05 Krauthgamer-Rabani rsquo06 Devanur-Khot-Saket-Vishnoi lsquo06
bull Disproves the GLARV Conjecture bull Previous best lower bound 116 [Zatloukal rsquo04]
Integrality Gap Lower Bound
Thm Construct graph G(1hellipnE) and unit vector assignment i -gt vi є Rn st
1 G is an ldquoexpanderrdquo every frac14 balanced cut has Ω(|E|(log log n)log n) edges (Kahn-Kalai-Linial)
2 ldquoLowrdquo SDP solution O(|E|log n)
3 Well-Separatedness Σiltj ||vi-vj||2 = n2
4 Triangle inequality d(ij)=||vi-vj||2 is a metric
Integrality gap Ω(log log n)
Starting Point Hypercube()
H=-11k
n = 2k
(111)
(-111)
(11-1)
(1-11)
(1-1-1)
(-1-11)
(-11-1) (-1-1-1)
Hypercube hellip H=-11k Advantages
bull Understand cuts in H tools from Fourier Analysis
bull Vertex is a vector in Rk starting point for SDP solution
But hellip
bull Hypercube has ldquosmallrdquo balanced cuts coordinate cuts have 1k fraction of the edges
Cuts in Hypercube Coordinate of edges = |E(H)|k
Edges across pairs of vertices differing in i-th bit
Cuts in Hypercube hellip
decompose into coordinate cuts
any balanced cut has a coord cut which contributes E(H)k2 edges
Kahn-Kalai-Linial
any balanced cut has a coordinate cut which contributes E(H) (log
k)k2 edges
Increasing Size of Balanced Cuts
consider balanced cuts in which coordinates are indistinguishable
(wrt to their contribution to the cut)
can be achieved by symmetrizing the hypercube
each coordinate contributes equally total E(H) (log k)k edges
eg 4-dim hypercube
(1111)
(-1-1-1-1)
(111-1) (-1111) (1-111) (11-11)
(-1-1-11) (1-1-1-1) (-11-1-1) (-1-11-1)
(11-1-1) (-111-1) (-1-111) (1-1-11)
(1-11-1) (-11-11)
More Formally hellipH=-11k with a rotation group acting on its coordinates
Partitions H into equivalence classes V1hellipVn
Each Vi is a vertex Edges are hypercube edges
G(VE) |E(G)|=|E(H)| k ~ log n
Balanced cuts in G correspond to balanced cuts in H
KKL any balanced cut (in H) has ldquoardquo coordinate cut which contributes E(H) (log k)k2 edges to the cut
Group is transitive ldquoeveryrdquo coordinate cut has the same contribution
Any balanced cut in G has ge E(G) (log k)k = E(G) (log log n)(log n)
Integrality Gap Lower Bound
Thm Construct graph G(1hellipnE) and unit vector assignment i -gt vi є Rn st
1 G is an ldquoexpanderrdquo every frac14 balanced cut has Ω(|E|(log log n)log n) edges (Kahn-Kalai-Linial)
2 ldquoLowrdquo SDP solution O(|E|log n)
3 Well-Separatedness Σiltj ||vi-vj||2 = n2
4 Triangle inequality d(ij)=||vi-vj||2 is a metric
Integrality gap Ω(log log n)
SDP Solution
(1111)
(-1-1-1-1)
(111-1) (-1111) (1-111) (11-11)
(-1-1-11) (1-1-1-1) (-11-1-1) (-1-11-1)
(11-1-1) (-111-1) (-1-111) (1-1-11)
(1-11-1) (-11-11)
Formally SDP SolutionVertex equivalence class x1 x2 hellip xk (rotations)
Vector (1radick) Σj xj
Observations
bull Edge across two nodes differing in one bit
Contribution to sdp ~ 1k sdp le |E(G)|k = |E(G)|(log n)
Triangle Inequality (little bit of work and case analysis)
For most classes x1 hellip xk is ldquonearly orthogonalrdquo
Hidden Gram-Schmidt Orthogonalization Tensoring Well-separatedness
Conclusion
bull (Simple) log log n integrality gap for SCBS
bull Close the gap between log log n and log n
bull [Lee-Naor rsquo06 Cheeger-Kleiner rsquo06] Another counter-example May give (log n)c
Thank you
- Slide 1
- CS Story Sparsity
- Applications
- Estimating Sparsity
- History
- Outline of this Talk
- Slide 7
- Quadratic Program for BS
- SDP for Balanced Separator
- Why is this a Relaxation
- Quadratic Program for BS hellip
- SDP for Balanced Separator hellip
- Geometry of Triangle Ineq
- Integrality Gap Upper Bound
- Slide 15
- Math Story Metric Embeddings
- Negative Type Metrics (squared-L2)
- Embedding NEG into L1
- Cuts and L1 Metrics
- Sparsest Cut asymp Optimizing Over L1
- Metric Embeddings amp Sparsity
- Integrality Gap Lower Bound
- Slide 23
- Slide 24
- Recall Integrality Gap Lower Bound
- Slide 26
- Starting Point Hypercube()
- Hypercube hellip
- Cuts in Hypercube Coordinate
- Cuts in Hypercube hellip
- Kahn-Kalai-Linial
- Increasing Size of Balanced Cuts
- eg 4-dim hypercube
- More Formally hellip
- Slide 35
- SDP Solution
- Formally SDP Solution
- Conclusion
-
HistoryAlgorithms
Spectral Graph partitioning Alon-Milman rsquo85 Speilman-Teng rsquo96(eigenvector based)
O(log n) Leighton-Rao rsquo88 (Linear Programming (LP) based)
O(log n) London-Linial-Rabinovitch rsquo94 Aumann-Rabani rsquo94 (connection to metric embeddings)
O(radiclog n) Arora-Rao-Vazirani lsquo04(Semi-Definite Programming (SDP) based)
Hardness NP-Hard Hard to approximate within any constant factor (assuming UGC)
Chawla-Krauthgamer-Kumar-Rabani-Sivakumar lsquo05 Khot-Vishnoi lsquo05
Outline of this Talk Part I
1 Graph PartitioningMotivation amp HistorySDP Approach
2 Embedding Negative Type Metrics into L1 Metric Spaces and EmbeddabilityCut Cone asymp L1
Negative Type Metrics as SDP SolutionsLLRAR Connection
Part II 1 Integrality Gap instance
Hypercube and Cutsbull Kahn-Kalai-Linial (isoperimetry of hypercube)
The GraphThe SDP Solution
2 Conclusion
Outline of this Talk Part I
1 Graph PartitioningMotivation amp HistorySDP Approach
2 Embedding Negative Type Metrics into L1 Metric Spaces and EmbeddabilityCut Cone asymp L1
Negative Type Metrics as SDP SolutionsLLRAR Connection Part II
1 Integrality Gap instance Hypercube and Cuts
bull Kahn-Kalai-Linial (isoperimetry of hypercube)The GraphThe SDP Solution
2 Conclusion
Quadratic Program for BS
i vi -11
Minimize frac14 |vi - vj |2
ij ε E
|vi - vj |2 = n2
iltj
Quadratic ProgramInput G(VE)
Output (SSc)
st
|S||Sc| = n2
which minimizes
|E(SSc)|
Balanced Separator
SDP for Balanced Separator
i vi -11
Minimize frac14 |vi - vj |2
ij ε E
|vi - vj |2 = n2
iltj
Quadratic Program SDP Relaxation
i vi Rn ||vi||=1
Minimize frac14 || vi - vj ||2
ij E
Well-Separatedness
|| vi - vj ||2 = n2
iltj
Why is this a Relaxation SDP Relaxation
G(VE)
i vi Rn ||vi||=1
Minimize frac14 || vi - vj ||2
ij E
Well-Separatedness
|| vi - vj ||2 = n2
iltj
Relaxation
u be a unit vector and (SSc)|S| |Sc| = n2
For i є S vi = u i є Sc vi = - u ||vi-vj||2 = 4 δS(ij) Cost of solution = |E(SSc)| sdp le opt
SDP can be computed in polynomial time
Boils down to the spectral approach Nothing gained()
Quadratic Program for BS hellip
i vi -11
Minimize frac14 |vi - vj |2
ij ε E
|vi - vj |2 = n2
iltj
Triangle Inequality i j k |vi - vj |2
+ |vj - vk |2 |vi - vk|2 (redundant)
Quadratic Program
Input G(VE)
Output (SSc)
st
|S||Sc| = n2
which minimizes
|E(SSc)|
Balanced Separator
SDP for Balanced Separator hellip
i vi -11
Minimize frac14 |vi - vj |2
ij ε E
|vi - vj |2 = n2
iltj
Triangle Inequality i j k |vi - vj |2+|vj - vk |2|vi - vk|2
Quadratic Program SDP Relaxation
i vi Rn ||vi||=1
Minimize frac14 || vi - vj ||2
ij E
Well-Separatedness
|| vi - vj ||2 = n2
iltj
Triangle Inequality i j k ||vi - vj||2 +||vj - vk||2 ||vi - vk||2
Still a relaxation hellip
Geometry of Triangle Ineq
le 90ovi
vk
vj
each step of length 1
t-steps length at-most radict
Rules out the embedding obtained by the spectral method
Integrality Gap Upper Bound
Arora-Rao-Vazirani rsquo04
O(radiclog n) for Sparsest Cut Balanced Separator
sdp within a factor of O(radiclog n) of the opt
Integrality gap max over all graphs on n vertices the ratio of
optsdp (as a function of n)
ARV conjectured that the integrality gap is upper bounded by some constant (independent of n)
Lack of any counterexample
Outline of this Talk Part I
1 Graph PartitioningMotivation amp HistorySDP Approach
2 Embedding Negative Type Metrics into L1 Metric Spaces and EmbeddabilityCut Cone asymp L1
Negative Type Metrics as SDP SolutionsLLRAR Connection
Part II 1 Integrality Gap instance
Hypercube and Cutsbull Kahn-Kalai-Linial (isoperimetry of hypercube)
The GraphThe SDP Solution
2 Conclusion
Math Story Metric Embeddings
bull Metric is a distance function d on [n] x [n] st d(i j) + d(j k) d(i k)
(triangle inequality)
bull Metric d embeds into metric with distortion 1 if there is a map φ st
ij d(i j) (φ(i) φ(j)) d(i j) (distances are preserved upto a factor of )
Negative Type Metrics (squared-L2)
bull d on 12hellipn is of negative type if there are vectors v1 v2 hellip vn
bull Such that d(i j) = || vi - vj ||2 satisfies triangle inequality
bull Same as i j k || vi - vj ||2
+ || vj - vk ||2 || vi - v k||2
bull NEG = class of such metrics bull arise as SDP solutions
bull L1 NEG
Embedding NEG into L1
Conjecture (Explicit by Goemans Linial abt rsquo95)
Every NEG metric embeds into L1 with
O(1) (constant) distortion
Whatrsquos the connection to sparsity
Cuts and L1 Metrics
pS non-negative real for every subset S of [n]
d(ij) = pSδS(ij)
Fact d is isometrically embeddable in L1
Further Every L1 embeddable metric on n-points can be written as non-negative linear sum of cut-metrics on 1hellipn
Cut-Metrics on 12hellipn
δS(i j) = 1 if i j are separated by (SSc) = 0 otherwise
S Sc
Sparsest Cut asymp Optimizing Over L1
[Aumann Rabani 98 Linial London Rabinovich rsquo94]
bull LP-relaxation over all METRICSbull [Bourgain rsquo85] Every n-point metric embeds
into L1 with O(log n) distortionbull O(log n) factor approximation for Sparsity
i~j δS(i j)
---------- iltj δS(i j)
Minimize S V
Minimize d is L1
i~j d(i j)
---------- iiltj d(i j)
=
Metric Embeddings amp Sparsity
bull Optimizing over cuts asymp Optimizing over L1 metrics
bull SDP solution asymp Optimizing over NEG
bull Goemans-LinialARV Conjecture NEG embeds in L1 with O(1) distortion Integrality Gap is O(1)
bull Implies O(1) approx algo for estimating sparsity
Integrality Gap Lower Bound
Sparsest Cut Balanced Separator
log log n integrality gap instance
Khot-Vishnoi rsquo05 Krauthgamer-Rabani lsquo06Devanur-Khot-Saket-Vishnoi lsquo06
bull Disproves the GLARV Conjecture bull Previous best lower bound 116 [Zatloukal rsquo04]
Outline of this Talk Part I
1 Graph PartitioningMotivation amp HistorySDP Approach
2 Embedding Negative Type Metrics into L1 Metric Spaces and EmbeddabilityCut Cone asymp L1
Negative Type Metrics as SDP SolutionsLLRAR Connection Part II
1 Integrality Gap instance Hypercube and Cuts
bull Kahn-Kalai-Linial (isoperimetry of hypercube)The GraphThe SDP Solution
2 Conclusion
Outline of this Talk Part I
1 Graph PartitioningMotivation amp HistorySDP Approach
2 Embedding Negative Type Metrics into L1 Metric Spaces and EmbeddabilityCut Cone asymp L1
Negative Type Metrics as SDP SolutionsLLRAR Connection Part II
1 Integrality Gap instance Hypercube and Cuts
bull Kahn-Kalai-Linial (isoperimetry of hypercube)The GraphThe SDP Solution
2 Conclusion
Recall Integrality Gap Lower Bound
Sparsest Cut Balanced Separator
log log n integrality gap instance
Khot-Vishnoi rsquo05 Krauthgamer-Rabani rsquo06 Devanur-Khot-Saket-Vishnoi lsquo06
bull Disproves the GLARV Conjecture bull Previous best lower bound 116 [Zatloukal rsquo04]
Integrality Gap Lower Bound
Thm Construct graph G(1hellipnE) and unit vector assignment i -gt vi є Rn st
1 G is an ldquoexpanderrdquo every frac14 balanced cut has Ω(|E|(log log n)log n) edges (Kahn-Kalai-Linial)
2 ldquoLowrdquo SDP solution O(|E|log n)
3 Well-Separatedness Σiltj ||vi-vj||2 = n2
4 Triangle inequality d(ij)=||vi-vj||2 is a metric
Integrality gap Ω(log log n)
Starting Point Hypercube()
H=-11k
n = 2k
(111)
(-111)
(11-1)
(1-11)
(1-1-1)
(-1-11)
(-11-1) (-1-1-1)
Hypercube hellip H=-11k Advantages
bull Understand cuts in H tools from Fourier Analysis
bull Vertex is a vector in Rk starting point for SDP solution
But hellip
bull Hypercube has ldquosmallrdquo balanced cuts coordinate cuts have 1k fraction of the edges
Cuts in Hypercube Coordinate of edges = |E(H)|k
Edges across pairs of vertices differing in i-th bit
Cuts in Hypercube hellip
decompose into coordinate cuts
any balanced cut has a coord cut which contributes E(H)k2 edges
Kahn-Kalai-Linial
any balanced cut has a coordinate cut which contributes E(H) (log
k)k2 edges
Increasing Size of Balanced Cuts
consider balanced cuts in which coordinates are indistinguishable
(wrt to their contribution to the cut)
can be achieved by symmetrizing the hypercube
each coordinate contributes equally total E(H) (log k)k edges
eg 4-dim hypercube
(1111)
(-1-1-1-1)
(111-1) (-1111) (1-111) (11-11)
(-1-1-11) (1-1-1-1) (-11-1-1) (-1-11-1)
(11-1-1) (-111-1) (-1-111) (1-1-11)
(1-11-1) (-11-11)
More Formally hellipH=-11k with a rotation group acting on its coordinates
Partitions H into equivalence classes V1hellipVn
Each Vi is a vertex Edges are hypercube edges
G(VE) |E(G)|=|E(H)| k ~ log n
Balanced cuts in G correspond to balanced cuts in H
KKL any balanced cut (in H) has ldquoardquo coordinate cut which contributes E(H) (log k)k2 edges to the cut
Group is transitive ldquoeveryrdquo coordinate cut has the same contribution
Any balanced cut in G has ge E(G) (log k)k = E(G) (log log n)(log n)
Integrality Gap Lower Bound
Thm Construct graph G(1hellipnE) and unit vector assignment i -gt vi є Rn st
1 G is an ldquoexpanderrdquo every frac14 balanced cut has Ω(|E|(log log n)log n) edges (Kahn-Kalai-Linial)
2 ldquoLowrdquo SDP solution O(|E|log n)
3 Well-Separatedness Σiltj ||vi-vj||2 = n2
4 Triangle inequality d(ij)=||vi-vj||2 is a metric
Integrality gap Ω(log log n)
SDP Solution
(1111)
(-1-1-1-1)
(111-1) (-1111) (1-111) (11-11)
(-1-1-11) (1-1-1-1) (-11-1-1) (-1-11-1)
(11-1-1) (-111-1) (-1-111) (1-1-11)
(1-11-1) (-11-11)
Formally SDP SolutionVertex equivalence class x1 x2 hellip xk (rotations)
Vector (1radick) Σj xj
Observations
bull Edge across two nodes differing in one bit
Contribution to sdp ~ 1k sdp le |E(G)|k = |E(G)|(log n)
Triangle Inequality (little bit of work and case analysis)
For most classes x1 hellip xk is ldquonearly orthogonalrdquo
Hidden Gram-Schmidt Orthogonalization Tensoring Well-separatedness
Conclusion
bull (Simple) log log n integrality gap for SCBS
bull Close the gap between log log n and log n
bull [Lee-Naor rsquo06 Cheeger-Kleiner rsquo06] Another counter-example May give (log n)c
Thank you
- Slide 1
- CS Story Sparsity
- Applications
- Estimating Sparsity
- History
- Outline of this Talk
- Slide 7
- Quadratic Program for BS
- SDP for Balanced Separator
- Why is this a Relaxation
- Quadratic Program for BS hellip
- SDP for Balanced Separator hellip
- Geometry of Triangle Ineq
- Integrality Gap Upper Bound
- Slide 15
- Math Story Metric Embeddings
- Negative Type Metrics (squared-L2)
- Embedding NEG into L1
- Cuts and L1 Metrics
- Sparsest Cut asymp Optimizing Over L1
- Metric Embeddings amp Sparsity
- Integrality Gap Lower Bound
- Slide 23
- Slide 24
- Recall Integrality Gap Lower Bound
- Slide 26
- Starting Point Hypercube()
- Hypercube hellip
- Cuts in Hypercube Coordinate
- Cuts in Hypercube hellip
- Kahn-Kalai-Linial
- Increasing Size of Balanced Cuts
- eg 4-dim hypercube
- More Formally hellip
- Slide 35
- SDP Solution
- Formally SDP Solution
- Conclusion
-
Outline of this Talk Part I
1 Graph PartitioningMotivation amp HistorySDP Approach
2 Embedding Negative Type Metrics into L1 Metric Spaces and EmbeddabilityCut Cone asymp L1
Negative Type Metrics as SDP SolutionsLLRAR Connection
Part II 1 Integrality Gap instance
Hypercube and Cutsbull Kahn-Kalai-Linial (isoperimetry of hypercube)
The GraphThe SDP Solution
2 Conclusion
Outline of this Talk Part I
1 Graph PartitioningMotivation amp HistorySDP Approach
2 Embedding Negative Type Metrics into L1 Metric Spaces and EmbeddabilityCut Cone asymp L1
Negative Type Metrics as SDP SolutionsLLRAR Connection Part II
1 Integrality Gap instance Hypercube and Cuts
bull Kahn-Kalai-Linial (isoperimetry of hypercube)The GraphThe SDP Solution
2 Conclusion
Quadratic Program for BS
i vi -11
Minimize frac14 |vi - vj |2
ij ε E
|vi - vj |2 = n2
iltj
Quadratic ProgramInput G(VE)
Output (SSc)
st
|S||Sc| = n2
which minimizes
|E(SSc)|
Balanced Separator
SDP for Balanced Separator
i vi -11
Minimize frac14 |vi - vj |2
ij ε E
|vi - vj |2 = n2
iltj
Quadratic Program SDP Relaxation
i vi Rn ||vi||=1
Minimize frac14 || vi - vj ||2
ij E
Well-Separatedness
|| vi - vj ||2 = n2
iltj
Why is this a Relaxation SDP Relaxation
G(VE)
i vi Rn ||vi||=1
Minimize frac14 || vi - vj ||2
ij E
Well-Separatedness
|| vi - vj ||2 = n2
iltj
Relaxation
u be a unit vector and (SSc)|S| |Sc| = n2
For i є S vi = u i є Sc vi = - u ||vi-vj||2 = 4 δS(ij) Cost of solution = |E(SSc)| sdp le opt
SDP can be computed in polynomial time
Boils down to the spectral approach Nothing gained()
Quadratic Program for BS hellip
i vi -11
Minimize frac14 |vi - vj |2
ij ε E
|vi - vj |2 = n2
iltj
Triangle Inequality i j k |vi - vj |2
+ |vj - vk |2 |vi - vk|2 (redundant)
Quadratic Program
Input G(VE)
Output (SSc)
st
|S||Sc| = n2
which minimizes
|E(SSc)|
Balanced Separator
SDP for Balanced Separator hellip
i vi -11
Minimize frac14 |vi - vj |2
ij ε E
|vi - vj |2 = n2
iltj
Triangle Inequality i j k |vi - vj |2+|vj - vk |2|vi - vk|2
Quadratic Program SDP Relaxation
i vi Rn ||vi||=1
Minimize frac14 || vi - vj ||2
ij E
Well-Separatedness
|| vi - vj ||2 = n2
iltj
Triangle Inequality i j k ||vi - vj||2 +||vj - vk||2 ||vi - vk||2
Still a relaxation hellip
Geometry of Triangle Ineq
le 90ovi
vk
vj
each step of length 1
t-steps length at-most radict
Rules out the embedding obtained by the spectral method
Integrality Gap Upper Bound
Arora-Rao-Vazirani rsquo04
O(radiclog n) for Sparsest Cut Balanced Separator
sdp within a factor of O(radiclog n) of the opt
Integrality gap max over all graphs on n vertices the ratio of
optsdp (as a function of n)
ARV conjectured that the integrality gap is upper bounded by some constant (independent of n)
Lack of any counterexample
Outline of this Talk Part I
1 Graph PartitioningMotivation amp HistorySDP Approach
2 Embedding Negative Type Metrics into L1 Metric Spaces and EmbeddabilityCut Cone asymp L1
Negative Type Metrics as SDP SolutionsLLRAR Connection
Part II 1 Integrality Gap instance
Hypercube and Cutsbull Kahn-Kalai-Linial (isoperimetry of hypercube)
The GraphThe SDP Solution
2 Conclusion
Math Story Metric Embeddings
bull Metric is a distance function d on [n] x [n] st d(i j) + d(j k) d(i k)
(triangle inequality)
bull Metric d embeds into metric with distortion 1 if there is a map φ st
ij d(i j) (φ(i) φ(j)) d(i j) (distances are preserved upto a factor of )
Negative Type Metrics (squared-L2)
bull d on 12hellipn is of negative type if there are vectors v1 v2 hellip vn
bull Such that d(i j) = || vi - vj ||2 satisfies triangle inequality
bull Same as i j k || vi - vj ||2
+ || vj - vk ||2 || vi - v k||2
bull NEG = class of such metrics bull arise as SDP solutions
bull L1 NEG
Embedding NEG into L1
Conjecture (Explicit by Goemans Linial abt rsquo95)
Every NEG metric embeds into L1 with
O(1) (constant) distortion
Whatrsquos the connection to sparsity
Cuts and L1 Metrics
pS non-negative real for every subset S of [n]
d(ij) = pSδS(ij)
Fact d is isometrically embeddable in L1
Further Every L1 embeddable metric on n-points can be written as non-negative linear sum of cut-metrics on 1hellipn
Cut-Metrics on 12hellipn
δS(i j) = 1 if i j are separated by (SSc) = 0 otherwise
S Sc
Sparsest Cut asymp Optimizing Over L1
[Aumann Rabani 98 Linial London Rabinovich rsquo94]
bull LP-relaxation over all METRICSbull [Bourgain rsquo85] Every n-point metric embeds
into L1 with O(log n) distortionbull O(log n) factor approximation for Sparsity
i~j δS(i j)
---------- iltj δS(i j)
Minimize S V
Minimize d is L1
i~j d(i j)
---------- iiltj d(i j)
=
Metric Embeddings amp Sparsity
bull Optimizing over cuts asymp Optimizing over L1 metrics
bull SDP solution asymp Optimizing over NEG
bull Goemans-LinialARV Conjecture NEG embeds in L1 with O(1) distortion Integrality Gap is O(1)
bull Implies O(1) approx algo for estimating sparsity
Integrality Gap Lower Bound
Sparsest Cut Balanced Separator
log log n integrality gap instance
Khot-Vishnoi rsquo05 Krauthgamer-Rabani lsquo06Devanur-Khot-Saket-Vishnoi lsquo06
bull Disproves the GLARV Conjecture bull Previous best lower bound 116 [Zatloukal rsquo04]
Outline of this Talk Part I
1 Graph PartitioningMotivation amp HistorySDP Approach
2 Embedding Negative Type Metrics into L1 Metric Spaces and EmbeddabilityCut Cone asymp L1
Negative Type Metrics as SDP SolutionsLLRAR Connection Part II
1 Integrality Gap instance Hypercube and Cuts
bull Kahn-Kalai-Linial (isoperimetry of hypercube)The GraphThe SDP Solution
2 Conclusion
Outline of this Talk Part I
1 Graph PartitioningMotivation amp HistorySDP Approach
2 Embedding Negative Type Metrics into L1 Metric Spaces and EmbeddabilityCut Cone asymp L1
Negative Type Metrics as SDP SolutionsLLRAR Connection Part II
1 Integrality Gap instance Hypercube and Cuts
bull Kahn-Kalai-Linial (isoperimetry of hypercube)The GraphThe SDP Solution
2 Conclusion
Recall Integrality Gap Lower Bound
Sparsest Cut Balanced Separator
log log n integrality gap instance
Khot-Vishnoi rsquo05 Krauthgamer-Rabani rsquo06 Devanur-Khot-Saket-Vishnoi lsquo06
bull Disproves the GLARV Conjecture bull Previous best lower bound 116 [Zatloukal rsquo04]
Integrality Gap Lower Bound
Thm Construct graph G(1hellipnE) and unit vector assignment i -gt vi є Rn st
1 G is an ldquoexpanderrdquo every frac14 balanced cut has Ω(|E|(log log n)log n) edges (Kahn-Kalai-Linial)
2 ldquoLowrdquo SDP solution O(|E|log n)
3 Well-Separatedness Σiltj ||vi-vj||2 = n2
4 Triangle inequality d(ij)=||vi-vj||2 is a metric
Integrality gap Ω(log log n)
Starting Point Hypercube()
H=-11k
n = 2k
(111)
(-111)
(11-1)
(1-11)
(1-1-1)
(-1-11)
(-11-1) (-1-1-1)
Hypercube hellip H=-11k Advantages
bull Understand cuts in H tools from Fourier Analysis
bull Vertex is a vector in Rk starting point for SDP solution
But hellip
bull Hypercube has ldquosmallrdquo balanced cuts coordinate cuts have 1k fraction of the edges
Cuts in Hypercube Coordinate of edges = |E(H)|k
Edges across pairs of vertices differing in i-th bit
Cuts in Hypercube hellip
decompose into coordinate cuts
any balanced cut has a coord cut which contributes E(H)k2 edges
Kahn-Kalai-Linial
any balanced cut has a coordinate cut which contributes E(H) (log
k)k2 edges
Increasing Size of Balanced Cuts
consider balanced cuts in which coordinates are indistinguishable
(wrt to their contribution to the cut)
can be achieved by symmetrizing the hypercube
each coordinate contributes equally total E(H) (log k)k edges
eg 4-dim hypercube
(1111)
(-1-1-1-1)
(111-1) (-1111) (1-111) (11-11)
(-1-1-11) (1-1-1-1) (-11-1-1) (-1-11-1)
(11-1-1) (-111-1) (-1-111) (1-1-11)
(1-11-1) (-11-11)
More Formally hellipH=-11k with a rotation group acting on its coordinates
Partitions H into equivalence classes V1hellipVn
Each Vi is a vertex Edges are hypercube edges
G(VE) |E(G)|=|E(H)| k ~ log n
Balanced cuts in G correspond to balanced cuts in H
KKL any balanced cut (in H) has ldquoardquo coordinate cut which contributes E(H) (log k)k2 edges to the cut
Group is transitive ldquoeveryrdquo coordinate cut has the same contribution
Any balanced cut in G has ge E(G) (log k)k = E(G) (log log n)(log n)
Integrality Gap Lower Bound
Thm Construct graph G(1hellipnE) and unit vector assignment i -gt vi є Rn st
1 G is an ldquoexpanderrdquo every frac14 balanced cut has Ω(|E|(log log n)log n) edges (Kahn-Kalai-Linial)
2 ldquoLowrdquo SDP solution O(|E|log n)
3 Well-Separatedness Σiltj ||vi-vj||2 = n2
4 Triangle inequality d(ij)=||vi-vj||2 is a metric
Integrality gap Ω(log log n)
SDP Solution
(1111)
(-1-1-1-1)
(111-1) (-1111) (1-111) (11-11)
(-1-1-11) (1-1-1-1) (-11-1-1) (-1-11-1)
(11-1-1) (-111-1) (-1-111) (1-1-11)
(1-11-1) (-11-11)
Formally SDP SolutionVertex equivalence class x1 x2 hellip xk (rotations)
Vector (1radick) Σj xj
Observations
bull Edge across two nodes differing in one bit
Contribution to sdp ~ 1k sdp le |E(G)|k = |E(G)|(log n)
Triangle Inequality (little bit of work and case analysis)
For most classes x1 hellip xk is ldquonearly orthogonalrdquo
Hidden Gram-Schmidt Orthogonalization Tensoring Well-separatedness
Conclusion
bull (Simple) log log n integrality gap for SCBS
bull Close the gap between log log n and log n
bull [Lee-Naor rsquo06 Cheeger-Kleiner rsquo06] Another counter-example May give (log n)c
Thank you
- Slide 1
- CS Story Sparsity
- Applications
- Estimating Sparsity
- History
- Outline of this Talk
- Slide 7
- Quadratic Program for BS
- SDP for Balanced Separator
- Why is this a Relaxation
- Quadratic Program for BS hellip
- SDP for Balanced Separator hellip
- Geometry of Triangle Ineq
- Integrality Gap Upper Bound
- Slide 15
- Math Story Metric Embeddings
- Negative Type Metrics (squared-L2)
- Embedding NEG into L1
- Cuts and L1 Metrics
- Sparsest Cut asymp Optimizing Over L1
- Metric Embeddings amp Sparsity
- Integrality Gap Lower Bound
- Slide 23
- Slide 24
- Recall Integrality Gap Lower Bound
- Slide 26
- Starting Point Hypercube()
- Hypercube hellip
- Cuts in Hypercube Coordinate
- Cuts in Hypercube hellip
- Kahn-Kalai-Linial
- Increasing Size of Balanced Cuts
- eg 4-dim hypercube
- More Formally hellip
- Slide 35
- SDP Solution
- Formally SDP Solution
- Conclusion
-
Outline of this Talk Part I
1 Graph PartitioningMotivation amp HistorySDP Approach
2 Embedding Negative Type Metrics into L1 Metric Spaces and EmbeddabilityCut Cone asymp L1
Negative Type Metrics as SDP SolutionsLLRAR Connection Part II
1 Integrality Gap instance Hypercube and Cuts
bull Kahn-Kalai-Linial (isoperimetry of hypercube)The GraphThe SDP Solution
2 Conclusion
Quadratic Program for BS
i vi -11
Minimize frac14 |vi - vj |2
ij ε E
|vi - vj |2 = n2
iltj
Quadratic ProgramInput G(VE)
Output (SSc)
st
|S||Sc| = n2
which minimizes
|E(SSc)|
Balanced Separator
SDP for Balanced Separator
i vi -11
Minimize frac14 |vi - vj |2
ij ε E
|vi - vj |2 = n2
iltj
Quadratic Program SDP Relaxation
i vi Rn ||vi||=1
Minimize frac14 || vi - vj ||2
ij E
Well-Separatedness
|| vi - vj ||2 = n2
iltj
Why is this a Relaxation SDP Relaxation
G(VE)
i vi Rn ||vi||=1
Minimize frac14 || vi - vj ||2
ij E
Well-Separatedness
|| vi - vj ||2 = n2
iltj
Relaxation
u be a unit vector and (SSc)|S| |Sc| = n2
For i є S vi = u i є Sc vi = - u ||vi-vj||2 = 4 δS(ij) Cost of solution = |E(SSc)| sdp le opt
SDP can be computed in polynomial time
Boils down to the spectral approach Nothing gained()
Quadratic Program for BS hellip
i vi -11
Minimize frac14 |vi - vj |2
ij ε E
|vi - vj |2 = n2
iltj
Triangle Inequality i j k |vi - vj |2
+ |vj - vk |2 |vi - vk|2 (redundant)
Quadratic Program
Input G(VE)
Output (SSc)
st
|S||Sc| = n2
which minimizes
|E(SSc)|
Balanced Separator
SDP for Balanced Separator hellip
i vi -11
Minimize frac14 |vi - vj |2
ij ε E
|vi - vj |2 = n2
iltj
Triangle Inequality i j k |vi - vj |2+|vj - vk |2|vi - vk|2
Quadratic Program SDP Relaxation
i vi Rn ||vi||=1
Minimize frac14 || vi - vj ||2
ij E
Well-Separatedness
|| vi - vj ||2 = n2
iltj
Triangle Inequality i j k ||vi - vj||2 +||vj - vk||2 ||vi - vk||2
Still a relaxation hellip
Geometry of Triangle Ineq
le 90ovi
vk
vj
each step of length 1
t-steps length at-most radict
Rules out the embedding obtained by the spectral method
Integrality Gap Upper Bound
Arora-Rao-Vazirani rsquo04
O(radiclog n) for Sparsest Cut Balanced Separator
sdp within a factor of O(radiclog n) of the opt
Integrality gap max over all graphs on n vertices the ratio of
optsdp (as a function of n)
ARV conjectured that the integrality gap is upper bounded by some constant (independent of n)
Lack of any counterexample
Outline of this Talk Part I
1 Graph PartitioningMotivation amp HistorySDP Approach
2 Embedding Negative Type Metrics into L1 Metric Spaces and EmbeddabilityCut Cone asymp L1
Negative Type Metrics as SDP SolutionsLLRAR Connection
Part II 1 Integrality Gap instance
Hypercube and Cutsbull Kahn-Kalai-Linial (isoperimetry of hypercube)
The GraphThe SDP Solution
2 Conclusion
Math Story Metric Embeddings
bull Metric is a distance function d on [n] x [n] st d(i j) + d(j k) d(i k)
(triangle inequality)
bull Metric d embeds into metric with distortion 1 if there is a map φ st
ij d(i j) (φ(i) φ(j)) d(i j) (distances are preserved upto a factor of )
Negative Type Metrics (squared-L2)
bull d on 12hellipn is of negative type if there are vectors v1 v2 hellip vn
bull Such that d(i j) = || vi - vj ||2 satisfies triangle inequality
bull Same as i j k || vi - vj ||2
+ || vj - vk ||2 || vi - v k||2
bull NEG = class of such metrics bull arise as SDP solutions
bull L1 NEG
Embedding NEG into L1
Conjecture (Explicit by Goemans Linial abt rsquo95)
Every NEG metric embeds into L1 with
O(1) (constant) distortion
Whatrsquos the connection to sparsity
Cuts and L1 Metrics
pS non-negative real for every subset S of [n]
d(ij) = pSδS(ij)
Fact d is isometrically embeddable in L1
Further Every L1 embeddable metric on n-points can be written as non-negative linear sum of cut-metrics on 1hellipn
Cut-Metrics on 12hellipn
δS(i j) = 1 if i j are separated by (SSc) = 0 otherwise
S Sc
Sparsest Cut asymp Optimizing Over L1
[Aumann Rabani 98 Linial London Rabinovich rsquo94]
bull LP-relaxation over all METRICSbull [Bourgain rsquo85] Every n-point metric embeds
into L1 with O(log n) distortionbull O(log n) factor approximation for Sparsity
i~j δS(i j)
---------- iltj δS(i j)
Minimize S V
Minimize d is L1
i~j d(i j)
---------- iiltj d(i j)
=
Metric Embeddings amp Sparsity
bull Optimizing over cuts asymp Optimizing over L1 metrics
bull SDP solution asymp Optimizing over NEG
bull Goemans-LinialARV Conjecture NEG embeds in L1 with O(1) distortion Integrality Gap is O(1)
bull Implies O(1) approx algo for estimating sparsity
Integrality Gap Lower Bound
Sparsest Cut Balanced Separator
log log n integrality gap instance
Khot-Vishnoi rsquo05 Krauthgamer-Rabani lsquo06Devanur-Khot-Saket-Vishnoi lsquo06
bull Disproves the GLARV Conjecture bull Previous best lower bound 116 [Zatloukal rsquo04]
Outline of this Talk Part I
1 Graph PartitioningMotivation amp HistorySDP Approach
2 Embedding Negative Type Metrics into L1 Metric Spaces and EmbeddabilityCut Cone asymp L1
Negative Type Metrics as SDP SolutionsLLRAR Connection Part II
1 Integrality Gap instance Hypercube and Cuts
bull Kahn-Kalai-Linial (isoperimetry of hypercube)The GraphThe SDP Solution
2 Conclusion
Outline of this Talk Part I
1 Graph PartitioningMotivation amp HistorySDP Approach
2 Embedding Negative Type Metrics into L1 Metric Spaces and EmbeddabilityCut Cone asymp L1
Negative Type Metrics as SDP SolutionsLLRAR Connection Part II
1 Integrality Gap instance Hypercube and Cuts
bull Kahn-Kalai-Linial (isoperimetry of hypercube)The GraphThe SDP Solution
2 Conclusion
Recall Integrality Gap Lower Bound
Sparsest Cut Balanced Separator
log log n integrality gap instance
Khot-Vishnoi rsquo05 Krauthgamer-Rabani rsquo06 Devanur-Khot-Saket-Vishnoi lsquo06
bull Disproves the GLARV Conjecture bull Previous best lower bound 116 [Zatloukal rsquo04]
Integrality Gap Lower Bound
Thm Construct graph G(1hellipnE) and unit vector assignment i -gt vi є Rn st
1 G is an ldquoexpanderrdquo every frac14 balanced cut has Ω(|E|(log log n)log n) edges (Kahn-Kalai-Linial)
2 ldquoLowrdquo SDP solution O(|E|log n)
3 Well-Separatedness Σiltj ||vi-vj||2 = n2
4 Triangle inequality d(ij)=||vi-vj||2 is a metric
Integrality gap Ω(log log n)
Starting Point Hypercube()
H=-11k
n = 2k
(111)
(-111)
(11-1)
(1-11)
(1-1-1)
(-1-11)
(-11-1) (-1-1-1)
Hypercube hellip H=-11k Advantages
bull Understand cuts in H tools from Fourier Analysis
bull Vertex is a vector in Rk starting point for SDP solution
But hellip
bull Hypercube has ldquosmallrdquo balanced cuts coordinate cuts have 1k fraction of the edges
Cuts in Hypercube Coordinate of edges = |E(H)|k
Edges across pairs of vertices differing in i-th bit
Cuts in Hypercube hellip
decompose into coordinate cuts
any balanced cut has a coord cut which contributes E(H)k2 edges
Kahn-Kalai-Linial
any balanced cut has a coordinate cut which contributes E(H) (log
k)k2 edges
Increasing Size of Balanced Cuts
consider balanced cuts in which coordinates are indistinguishable
(wrt to their contribution to the cut)
can be achieved by symmetrizing the hypercube
each coordinate contributes equally total E(H) (log k)k edges
eg 4-dim hypercube
(1111)
(-1-1-1-1)
(111-1) (-1111) (1-111) (11-11)
(-1-1-11) (1-1-1-1) (-11-1-1) (-1-11-1)
(11-1-1) (-111-1) (-1-111) (1-1-11)
(1-11-1) (-11-11)
More Formally hellipH=-11k with a rotation group acting on its coordinates
Partitions H into equivalence classes V1hellipVn
Each Vi is a vertex Edges are hypercube edges
G(VE) |E(G)|=|E(H)| k ~ log n
Balanced cuts in G correspond to balanced cuts in H
KKL any balanced cut (in H) has ldquoardquo coordinate cut which contributes E(H) (log k)k2 edges to the cut
Group is transitive ldquoeveryrdquo coordinate cut has the same contribution
Any balanced cut in G has ge E(G) (log k)k = E(G) (log log n)(log n)
Integrality Gap Lower Bound
Thm Construct graph G(1hellipnE) and unit vector assignment i -gt vi є Rn st
1 G is an ldquoexpanderrdquo every frac14 balanced cut has Ω(|E|(log log n)log n) edges (Kahn-Kalai-Linial)
2 ldquoLowrdquo SDP solution O(|E|log n)
3 Well-Separatedness Σiltj ||vi-vj||2 = n2
4 Triangle inequality d(ij)=||vi-vj||2 is a metric
Integrality gap Ω(log log n)
SDP Solution
(1111)
(-1-1-1-1)
(111-1) (-1111) (1-111) (11-11)
(-1-1-11) (1-1-1-1) (-11-1-1) (-1-11-1)
(11-1-1) (-111-1) (-1-111) (1-1-11)
(1-11-1) (-11-11)
Formally SDP SolutionVertex equivalence class x1 x2 hellip xk (rotations)
Vector (1radick) Σj xj
Observations
bull Edge across two nodes differing in one bit
Contribution to sdp ~ 1k sdp le |E(G)|k = |E(G)|(log n)
Triangle Inequality (little bit of work and case analysis)
For most classes x1 hellip xk is ldquonearly orthogonalrdquo
Hidden Gram-Schmidt Orthogonalization Tensoring Well-separatedness
Conclusion
bull (Simple) log log n integrality gap for SCBS
bull Close the gap between log log n and log n
bull [Lee-Naor rsquo06 Cheeger-Kleiner rsquo06] Another counter-example May give (log n)c
Thank you
- Slide 1
- CS Story Sparsity
- Applications
- Estimating Sparsity
- History
- Outline of this Talk
- Slide 7
- Quadratic Program for BS
- SDP for Balanced Separator
- Why is this a Relaxation
- Quadratic Program for BS hellip
- SDP for Balanced Separator hellip
- Geometry of Triangle Ineq
- Integrality Gap Upper Bound
- Slide 15
- Math Story Metric Embeddings
- Negative Type Metrics (squared-L2)
- Embedding NEG into L1
- Cuts and L1 Metrics
- Sparsest Cut asymp Optimizing Over L1
- Metric Embeddings amp Sparsity
- Integrality Gap Lower Bound
- Slide 23
- Slide 24
- Recall Integrality Gap Lower Bound
- Slide 26
- Starting Point Hypercube()
- Hypercube hellip
- Cuts in Hypercube Coordinate
- Cuts in Hypercube hellip
- Kahn-Kalai-Linial
- Increasing Size of Balanced Cuts
- eg 4-dim hypercube
- More Formally hellip
- Slide 35
- SDP Solution
- Formally SDP Solution
- Conclusion
-
Quadratic Program for BS
i vi -11
Minimize frac14 |vi - vj |2
ij ε E
|vi - vj |2 = n2
iltj
Quadratic ProgramInput G(VE)
Output (SSc)
st
|S||Sc| = n2
which minimizes
|E(SSc)|
Balanced Separator
SDP for Balanced Separator
i vi -11
Minimize frac14 |vi - vj |2
ij ε E
|vi - vj |2 = n2
iltj
Quadratic Program SDP Relaxation
i vi Rn ||vi||=1
Minimize frac14 || vi - vj ||2
ij E
Well-Separatedness
|| vi - vj ||2 = n2
iltj
Why is this a Relaxation SDP Relaxation
G(VE)
i vi Rn ||vi||=1
Minimize frac14 || vi - vj ||2
ij E
Well-Separatedness
|| vi - vj ||2 = n2
iltj
Relaxation
u be a unit vector and (SSc)|S| |Sc| = n2
For i є S vi = u i є Sc vi = - u ||vi-vj||2 = 4 δS(ij) Cost of solution = |E(SSc)| sdp le opt
SDP can be computed in polynomial time
Boils down to the spectral approach Nothing gained()
Quadratic Program for BS hellip
i vi -11
Minimize frac14 |vi - vj |2
ij ε E
|vi - vj |2 = n2
iltj
Triangle Inequality i j k |vi - vj |2
+ |vj - vk |2 |vi - vk|2 (redundant)
Quadratic Program
Input G(VE)
Output (SSc)
st
|S||Sc| = n2
which minimizes
|E(SSc)|
Balanced Separator
SDP for Balanced Separator hellip
i vi -11
Minimize frac14 |vi - vj |2
ij ε E
|vi - vj |2 = n2
iltj
Triangle Inequality i j k |vi - vj |2+|vj - vk |2|vi - vk|2
Quadratic Program SDP Relaxation
i vi Rn ||vi||=1
Minimize frac14 || vi - vj ||2
ij E
Well-Separatedness
|| vi - vj ||2 = n2
iltj
Triangle Inequality i j k ||vi - vj||2 +||vj - vk||2 ||vi - vk||2
Still a relaxation hellip
Geometry of Triangle Ineq
le 90ovi
vk
vj
each step of length 1
t-steps length at-most radict
Rules out the embedding obtained by the spectral method
Integrality Gap Upper Bound
Arora-Rao-Vazirani rsquo04
O(radiclog n) for Sparsest Cut Balanced Separator
sdp within a factor of O(radiclog n) of the opt
Integrality gap max over all graphs on n vertices the ratio of
optsdp (as a function of n)
ARV conjectured that the integrality gap is upper bounded by some constant (independent of n)
Lack of any counterexample
Outline of this Talk Part I
1 Graph PartitioningMotivation amp HistorySDP Approach
2 Embedding Negative Type Metrics into L1 Metric Spaces and EmbeddabilityCut Cone asymp L1
Negative Type Metrics as SDP SolutionsLLRAR Connection
Part II 1 Integrality Gap instance
Hypercube and Cutsbull Kahn-Kalai-Linial (isoperimetry of hypercube)
The GraphThe SDP Solution
2 Conclusion
Math Story Metric Embeddings
bull Metric is a distance function d on [n] x [n] st d(i j) + d(j k) d(i k)
(triangle inequality)
bull Metric d embeds into metric with distortion 1 if there is a map φ st
ij d(i j) (φ(i) φ(j)) d(i j) (distances are preserved upto a factor of )
Negative Type Metrics (squared-L2)
bull d on 12hellipn is of negative type if there are vectors v1 v2 hellip vn
bull Such that d(i j) = || vi - vj ||2 satisfies triangle inequality
bull Same as i j k || vi - vj ||2
+ || vj - vk ||2 || vi - v k||2
bull NEG = class of such metrics bull arise as SDP solutions
bull L1 NEG
Embedding NEG into L1
Conjecture (Explicit by Goemans Linial abt rsquo95)
Every NEG metric embeds into L1 with
O(1) (constant) distortion
Whatrsquos the connection to sparsity
Cuts and L1 Metrics
pS non-negative real for every subset S of [n]
d(ij) = pSδS(ij)
Fact d is isometrically embeddable in L1
Further Every L1 embeddable metric on n-points can be written as non-negative linear sum of cut-metrics on 1hellipn
Cut-Metrics on 12hellipn
δS(i j) = 1 if i j are separated by (SSc) = 0 otherwise
S Sc
Sparsest Cut asymp Optimizing Over L1
[Aumann Rabani 98 Linial London Rabinovich rsquo94]
bull LP-relaxation over all METRICSbull [Bourgain rsquo85] Every n-point metric embeds
into L1 with O(log n) distortionbull O(log n) factor approximation for Sparsity
i~j δS(i j)
---------- iltj δS(i j)
Minimize S V
Minimize d is L1
i~j d(i j)
---------- iiltj d(i j)
=
Metric Embeddings amp Sparsity
bull Optimizing over cuts asymp Optimizing over L1 metrics
bull SDP solution asymp Optimizing over NEG
bull Goemans-LinialARV Conjecture NEG embeds in L1 with O(1) distortion Integrality Gap is O(1)
bull Implies O(1) approx algo for estimating sparsity
Integrality Gap Lower Bound
Sparsest Cut Balanced Separator
log log n integrality gap instance
Khot-Vishnoi rsquo05 Krauthgamer-Rabani lsquo06Devanur-Khot-Saket-Vishnoi lsquo06
bull Disproves the GLARV Conjecture bull Previous best lower bound 116 [Zatloukal rsquo04]
Outline of this Talk Part I
1 Graph PartitioningMotivation amp HistorySDP Approach
2 Embedding Negative Type Metrics into L1 Metric Spaces and EmbeddabilityCut Cone asymp L1
Negative Type Metrics as SDP SolutionsLLRAR Connection Part II
1 Integrality Gap instance Hypercube and Cuts
bull Kahn-Kalai-Linial (isoperimetry of hypercube)The GraphThe SDP Solution
2 Conclusion
Outline of this Talk Part I
1 Graph PartitioningMotivation amp HistorySDP Approach
2 Embedding Negative Type Metrics into L1 Metric Spaces and EmbeddabilityCut Cone asymp L1
Negative Type Metrics as SDP SolutionsLLRAR Connection Part II
1 Integrality Gap instance Hypercube and Cuts
bull Kahn-Kalai-Linial (isoperimetry of hypercube)The GraphThe SDP Solution
2 Conclusion
Recall Integrality Gap Lower Bound
Sparsest Cut Balanced Separator
log log n integrality gap instance
Khot-Vishnoi rsquo05 Krauthgamer-Rabani rsquo06 Devanur-Khot-Saket-Vishnoi lsquo06
bull Disproves the GLARV Conjecture bull Previous best lower bound 116 [Zatloukal rsquo04]
Integrality Gap Lower Bound
Thm Construct graph G(1hellipnE) and unit vector assignment i -gt vi є Rn st
1 G is an ldquoexpanderrdquo every frac14 balanced cut has Ω(|E|(log log n)log n) edges (Kahn-Kalai-Linial)
2 ldquoLowrdquo SDP solution O(|E|log n)
3 Well-Separatedness Σiltj ||vi-vj||2 = n2
4 Triangle inequality d(ij)=||vi-vj||2 is a metric
Integrality gap Ω(log log n)
Starting Point Hypercube()
H=-11k
n = 2k
(111)
(-111)
(11-1)
(1-11)
(1-1-1)
(-1-11)
(-11-1) (-1-1-1)
Hypercube hellip H=-11k Advantages
bull Understand cuts in H tools from Fourier Analysis
bull Vertex is a vector in Rk starting point for SDP solution
But hellip
bull Hypercube has ldquosmallrdquo balanced cuts coordinate cuts have 1k fraction of the edges
Cuts in Hypercube Coordinate of edges = |E(H)|k
Edges across pairs of vertices differing in i-th bit
Cuts in Hypercube hellip
decompose into coordinate cuts
any balanced cut has a coord cut which contributes E(H)k2 edges
Kahn-Kalai-Linial
any balanced cut has a coordinate cut which contributes E(H) (log
k)k2 edges
Increasing Size of Balanced Cuts
consider balanced cuts in which coordinates are indistinguishable
(wrt to their contribution to the cut)
can be achieved by symmetrizing the hypercube
each coordinate contributes equally total E(H) (log k)k edges
eg 4-dim hypercube
(1111)
(-1-1-1-1)
(111-1) (-1111) (1-111) (11-11)
(-1-1-11) (1-1-1-1) (-11-1-1) (-1-11-1)
(11-1-1) (-111-1) (-1-111) (1-1-11)
(1-11-1) (-11-11)
More Formally hellipH=-11k with a rotation group acting on its coordinates
Partitions H into equivalence classes V1hellipVn
Each Vi is a vertex Edges are hypercube edges
G(VE) |E(G)|=|E(H)| k ~ log n
Balanced cuts in G correspond to balanced cuts in H
KKL any balanced cut (in H) has ldquoardquo coordinate cut which contributes E(H) (log k)k2 edges to the cut
Group is transitive ldquoeveryrdquo coordinate cut has the same contribution
Any balanced cut in G has ge E(G) (log k)k = E(G) (log log n)(log n)
Integrality Gap Lower Bound
Thm Construct graph G(1hellipnE) and unit vector assignment i -gt vi є Rn st
1 G is an ldquoexpanderrdquo every frac14 balanced cut has Ω(|E|(log log n)log n) edges (Kahn-Kalai-Linial)
2 ldquoLowrdquo SDP solution O(|E|log n)
3 Well-Separatedness Σiltj ||vi-vj||2 = n2
4 Triangle inequality d(ij)=||vi-vj||2 is a metric
Integrality gap Ω(log log n)
SDP Solution
(1111)
(-1-1-1-1)
(111-1) (-1111) (1-111) (11-11)
(-1-1-11) (1-1-1-1) (-11-1-1) (-1-11-1)
(11-1-1) (-111-1) (-1-111) (1-1-11)
(1-11-1) (-11-11)
Formally SDP SolutionVertex equivalence class x1 x2 hellip xk (rotations)
Vector (1radick) Σj xj
Observations
bull Edge across two nodes differing in one bit
Contribution to sdp ~ 1k sdp le |E(G)|k = |E(G)|(log n)
Triangle Inequality (little bit of work and case analysis)
For most classes x1 hellip xk is ldquonearly orthogonalrdquo
Hidden Gram-Schmidt Orthogonalization Tensoring Well-separatedness
Conclusion
bull (Simple) log log n integrality gap for SCBS
bull Close the gap between log log n and log n
bull [Lee-Naor rsquo06 Cheeger-Kleiner rsquo06] Another counter-example May give (log n)c
Thank you
- Slide 1
- CS Story Sparsity
- Applications
- Estimating Sparsity
- History
- Outline of this Talk
- Slide 7
- Quadratic Program for BS
- SDP for Balanced Separator
- Why is this a Relaxation
- Quadratic Program for BS hellip
- SDP for Balanced Separator hellip
- Geometry of Triangle Ineq
- Integrality Gap Upper Bound
- Slide 15
- Math Story Metric Embeddings
- Negative Type Metrics (squared-L2)
- Embedding NEG into L1
- Cuts and L1 Metrics
- Sparsest Cut asymp Optimizing Over L1
- Metric Embeddings amp Sparsity
- Integrality Gap Lower Bound
- Slide 23
- Slide 24
- Recall Integrality Gap Lower Bound
- Slide 26
- Starting Point Hypercube()
- Hypercube hellip
- Cuts in Hypercube Coordinate
- Cuts in Hypercube hellip
- Kahn-Kalai-Linial
- Increasing Size of Balanced Cuts
- eg 4-dim hypercube
- More Formally hellip
- Slide 35
- SDP Solution
- Formally SDP Solution
- Conclusion
-
SDP for Balanced Separator
i vi -11
Minimize frac14 |vi - vj |2
ij ε E
|vi - vj |2 = n2
iltj
Quadratic Program SDP Relaxation
i vi Rn ||vi||=1
Minimize frac14 || vi - vj ||2
ij E
Well-Separatedness
|| vi - vj ||2 = n2
iltj
Why is this a Relaxation SDP Relaxation
G(VE)
i vi Rn ||vi||=1
Minimize frac14 || vi - vj ||2
ij E
Well-Separatedness
|| vi - vj ||2 = n2
iltj
Relaxation
u be a unit vector and (SSc)|S| |Sc| = n2
For i є S vi = u i є Sc vi = - u ||vi-vj||2 = 4 δS(ij) Cost of solution = |E(SSc)| sdp le opt
SDP can be computed in polynomial time
Boils down to the spectral approach Nothing gained()
Quadratic Program for BS hellip
i vi -11
Minimize frac14 |vi - vj |2
ij ε E
|vi - vj |2 = n2
iltj
Triangle Inequality i j k |vi - vj |2
+ |vj - vk |2 |vi - vk|2 (redundant)
Quadratic Program
Input G(VE)
Output (SSc)
st
|S||Sc| = n2
which minimizes
|E(SSc)|
Balanced Separator
SDP for Balanced Separator hellip
i vi -11
Minimize frac14 |vi - vj |2
ij ε E
|vi - vj |2 = n2
iltj
Triangle Inequality i j k |vi - vj |2+|vj - vk |2|vi - vk|2
Quadratic Program SDP Relaxation
i vi Rn ||vi||=1
Minimize frac14 || vi - vj ||2
ij E
Well-Separatedness
|| vi - vj ||2 = n2
iltj
Triangle Inequality i j k ||vi - vj||2 +||vj - vk||2 ||vi - vk||2
Still a relaxation hellip
Geometry of Triangle Ineq
le 90ovi
vk
vj
each step of length 1
t-steps length at-most radict
Rules out the embedding obtained by the spectral method
Integrality Gap Upper Bound
Arora-Rao-Vazirani rsquo04
O(radiclog n) for Sparsest Cut Balanced Separator
sdp within a factor of O(radiclog n) of the opt
Integrality gap max over all graphs on n vertices the ratio of
optsdp (as a function of n)
ARV conjectured that the integrality gap is upper bounded by some constant (independent of n)
Lack of any counterexample
Outline of this Talk Part I
1 Graph PartitioningMotivation amp HistorySDP Approach
2 Embedding Negative Type Metrics into L1 Metric Spaces and EmbeddabilityCut Cone asymp L1
Negative Type Metrics as SDP SolutionsLLRAR Connection
Part II 1 Integrality Gap instance
Hypercube and Cutsbull Kahn-Kalai-Linial (isoperimetry of hypercube)
The GraphThe SDP Solution
2 Conclusion
Math Story Metric Embeddings
bull Metric is a distance function d on [n] x [n] st d(i j) + d(j k) d(i k)
(triangle inequality)
bull Metric d embeds into metric with distortion 1 if there is a map φ st
ij d(i j) (φ(i) φ(j)) d(i j) (distances are preserved upto a factor of )
Negative Type Metrics (squared-L2)
bull d on 12hellipn is of negative type if there are vectors v1 v2 hellip vn
bull Such that d(i j) = || vi - vj ||2 satisfies triangle inequality
bull Same as i j k || vi - vj ||2
+ || vj - vk ||2 || vi - v k||2
bull NEG = class of such metrics bull arise as SDP solutions
bull L1 NEG
Embedding NEG into L1
Conjecture (Explicit by Goemans Linial abt rsquo95)
Every NEG metric embeds into L1 with
O(1) (constant) distortion
Whatrsquos the connection to sparsity
Cuts and L1 Metrics
pS non-negative real for every subset S of [n]
d(ij) = pSδS(ij)
Fact d is isometrically embeddable in L1
Further Every L1 embeddable metric on n-points can be written as non-negative linear sum of cut-metrics on 1hellipn
Cut-Metrics on 12hellipn
δS(i j) = 1 if i j are separated by (SSc) = 0 otherwise
S Sc
Sparsest Cut asymp Optimizing Over L1
[Aumann Rabani 98 Linial London Rabinovich rsquo94]
bull LP-relaxation over all METRICSbull [Bourgain rsquo85] Every n-point metric embeds
into L1 with O(log n) distortionbull O(log n) factor approximation for Sparsity
i~j δS(i j)
---------- iltj δS(i j)
Minimize S V
Minimize d is L1
i~j d(i j)
---------- iiltj d(i j)
=
Metric Embeddings amp Sparsity
bull Optimizing over cuts asymp Optimizing over L1 metrics
bull SDP solution asymp Optimizing over NEG
bull Goemans-LinialARV Conjecture NEG embeds in L1 with O(1) distortion Integrality Gap is O(1)
bull Implies O(1) approx algo for estimating sparsity
Integrality Gap Lower Bound
Sparsest Cut Balanced Separator
log log n integrality gap instance
Khot-Vishnoi rsquo05 Krauthgamer-Rabani lsquo06Devanur-Khot-Saket-Vishnoi lsquo06
bull Disproves the GLARV Conjecture bull Previous best lower bound 116 [Zatloukal rsquo04]
Outline of this Talk Part I
1 Graph PartitioningMotivation amp HistorySDP Approach
2 Embedding Negative Type Metrics into L1 Metric Spaces and EmbeddabilityCut Cone asymp L1
Negative Type Metrics as SDP SolutionsLLRAR Connection Part II
1 Integrality Gap instance Hypercube and Cuts
bull Kahn-Kalai-Linial (isoperimetry of hypercube)The GraphThe SDP Solution
2 Conclusion
Outline of this Talk Part I
1 Graph PartitioningMotivation amp HistorySDP Approach
2 Embedding Negative Type Metrics into L1 Metric Spaces and EmbeddabilityCut Cone asymp L1
Negative Type Metrics as SDP SolutionsLLRAR Connection Part II
1 Integrality Gap instance Hypercube and Cuts
bull Kahn-Kalai-Linial (isoperimetry of hypercube)The GraphThe SDP Solution
2 Conclusion
Recall Integrality Gap Lower Bound
Sparsest Cut Balanced Separator
log log n integrality gap instance
Khot-Vishnoi rsquo05 Krauthgamer-Rabani rsquo06 Devanur-Khot-Saket-Vishnoi lsquo06
bull Disproves the GLARV Conjecture bull Previous best lower bound 116 [Zatloukal rsquo04]
Integrality Gap Lower Bound
Thm Construct graph G(1hellipnE) and unit vector assignment i -gt vi є Rn st
1 G is an ldquoexpanderrdquo every frac14 balanced cut has Ω(|E|(log log n)log n) edges (Kahn-Kalai-Linial)
2 ldquoLowrdquo SDP solution O(|E|log n)
3 Well-Separatedness Σiltj ||vi-vj||2 = n2
4 Triangle inequality d(ij)=||vi-vj||2 is a metric
Integrality gap Ω(log log n)
Starting Point Hypercube()
H=-11k
n = 2k
(111)
(-111)
(11-1)
(1-11)
(1-1-1)
(-1-11)
(-11-1) (-1-1-1)
Hypercube hellip H=-11k Advantages
bull Understand cuts in H tools from Fourier Analysis
bull Vertex is a vector in Rk starting point for SDP solution
But hellip
bull Hypercube has ldquosmallrdquo balanced cuts coordinate cuts have 1k fraction of the edges
Cuts in Hypercube Coordinate of edges = |E(H)|k
Edges across pairs of vertices differing in i-th bit
Cuts in Hypercube hellip
decompose into coordinate cuts
any balanced cut has a coord cut which contributes E(H)k2 edges
Kahn-Kalai-Linial
any balanced cut has a coordinate cut which contributes E(H) (log
k)k2 edges
Increasing Size of Balanced Cuts
consider balanced cuts in which coordinates are indistinguishable
(wrt to their contribution to the cut)
can be achieved by symmetrizing the hypercube
each coordinate contributes equally total E(H) (log k)k edges
eg 4-dim hypercube
(1111)
(-1-1-1-1)
(111-1) (-1111) (1-111) (11-11)
(-1-1-11) (1-1-1-1) (-11-1-1) (-1-11-1)
(11-1-1) (-111-1) (-1-111) (1-1-11)
(1-11-1) (-11-11)
More Formally hellipH=-11k with a rotation group acting on its coordinates
Partitions H into equivalence classes V1hellipVn
Each Vi is a vertex Edges are hypercube edges
G(VE) |E(G)|=|E(H)| k ~ log n
Balanced cuts in G correspond to balanced cuts in H
KKL any balanced cut (in H) has ldquoardquo coordinate cut which contributes E(H) (log k)k2 edges to the cut
Group is transitive ldquoeveryrdquo coordinate cut has the same contribution
Any balanced cut in G has ge E(G) (log k)k = E(G) (log log n)(log n)
Integrality Gap Lower Bound
Thm Construct graph G(1hellipnE) and unit vector assignment i -gt vi є Rn st
1 G is an ldquoexpanderrdquo every frac14 balanced cut has Ω(|E|(log log n)log n) edges (Kahn-Kalai-Linial)
2 ldquoLowrdquo SDP solution O(|E|log n)
3 Well-Separatedness Σiltj ||vi-vj||2 = n2
4 Triangle inequality d(ij)=||vi-vj||2 is a metric
Integrality gap Ω(log log n)
SDP Solution
(1111)
(-1-1-1-1)
(111-1) (-1111) (1-111) (11-11)
(-1-1-11) (1-1-1-1) (-11-1-1) (-1-11-1)
(11-1-1) (-111-1) (-1-111) (1-1-11)
(1-11-1) (-11-11)
Formally SDP SolutionVertex equivalence class x1 x2 hellip xk (rotations)
Vector (1radick) Σj xj
Observations
bull Edge across two nodes differing in one bit
Contribution to sdp ~ 1k sdp le |E(G)|k = |E(G)|(log n)
Triangle Inequality (little bit of work and case analysis)
For most classes x1 hellip xk is ldquonearly orthogonalrdquo
Hidden Gram-Schmidt Orthogonalization Tensoring Well-separatedness
Conclusion
bull (Simple) log log n integrality gap for SCBS
bull Close the gap between log log n and log n
bull [Lee-Naor rsquo06 Cheeger-Kleiner rsquo06] Another counter-example May give (log n)c
Thank you
- Slide 1
- CS Story Sparsity
- Applications
- Estimating Sparsity
- History
- Outline of this Talk
- Slide 7
- Quadratic Program for BS
- SDP for Balanced Separator
- Why is this a Relaxation
- Quadratic Program for BS hellip
- SDP for Balanced Separator hellip
- Geometry of Triangle Ineq
- Integrality Gap Upper Bound
- Slide 15
- Math Story Metric Embeddings
- Negative Type Metrics (squared-L2)
- Embedding NEG into L1
- Cuts and L1 Metrics
- Sparsest Cut asymp Optimizing Over L1
- Metric Embeddings amp Sparsity
- Integrality Gap Lower Bound
- Slide 23
- Slide 24
- Recall Integrality Gap Lower Bound
- Slide 26
- Starting Point Hypercube()
- Hypercube hellip
- Cuts in Hypercube Coordinate
- Cuts in Hypercube hellip
- Kahn-Kalai-Linial
- Increasing Size of Balanced Cuts
- eg 4-dim hypercube
- More Formally hellip
- Slide 35
- SDP Solution
- Formally SDP Solution
- Conclusion
-
Why is this a Relaxation SDP Relaxation
G(VE)
i vi Rn ||vi||=1
Minimize frac14 || vi - vj ||2
ij E
Well-Separatedness
|| vi - vj ||2 = n2
iltj
Relaxation
u be a unit vector and (SSc)|S| |Sc| = n2
For i є S vi = u i є Sc vi = - u ||vi-vj||2 = 4 δS(ij) Cost of solution = |E(SSc)| sdp le opt
SDP can be computed in polynomial time
Boils down to the spectral approach Nothing gained()
Quadratic Program for BS hellip
i vi -11
Minimize frac14 |vi - vj |2
ij ε E
|vi - vj |2 = n2
iltj
Triangle Inequality i j k |vi - vj |2
+ |vj - vk |2 |vi - vk|2 (redundant)
Quadratic Program
Input G(VE)
Output (SSc)
st
|S||Sc| = n2
which minimizes
|E(SSc)|
Balanced Separator
SDP for Balanced Separator hellip
i vi -11
Minimize frac14 |vi - vj |2
ij ε E
|vi - vj |2 = n2
iltj
Triangle Inequality i j k |vi - vj |2+|vj - vk |2|vi - vk|2
Quadratic Program SDP Relaxation
i vi Rn ||vi||=1
Minimize frac14 || vi - vj ||2
ij E
Well-Separatedness
|| vi - vj ||2 = n2
iltj
Triangle Inequality i j k ||vi - vj||2 +||vj - vk||2 ||vi - vk||2
Still a relaxation hellip
Geometry of Triangle Ineq
le 90ovi
vk
vj
each step of length 1
t-steps length at-most radict
Rules out the embedding obtained by the spectral method
Integrality Gap Upper Bound
Arora-Rao-Vazirani rsquo04
O(radiclog n) for Sparsest Cut Balanced Separator
sdp within a factor of O(radiclog n) of the opt
Integrality gap max over all graphs on n vertices the ratio of
optsdp (as a function of n)
ARV conjectured that the integrality gap is upper bounded by some constant (independent of n)
Lack of any counterexample
Outline of this Talk Part I
1 Graph PartitioningMotivation amp HistorySDP Approach
2 Embedding Negative Type Metrics into L1 Metric Spaces and EmbeddabilityCut Cone asymp L1
Negative Type Metrics as SDP SolutionsLLRAR Connection
Part II 1 Integrality Gap instance
Hypercube and Cutsbull Kahn-Kalai-Linial (isoperimetry of hypercube)
The GraphThe SDP Solution
2 Conclusion
Math Story Metric Embeddings
bull Metric is a distance function d on [n] x [n] st d(i j) + d(j k) d(i k)
(triangle inequality)
bull Metric d embeds into metric with distortion 1 if there is a map φ st
ij d(i j) (φ(i) φ(j)) d(i j) (distances are preserved upto a factor of )
Negative Type Metrics (squared-L2)
bull d on 12hellipn is of negative type if there are vectors v1 v2 hellip vn
bull Such that d(i j) = || vi - vj ||2 satisfies triangle inequality
bull Same as i j k || vi - vj ||2
+ || vj - vk ||2 || vi - v k||2
bull NEG = class of such metrics bull arise as SDP solutions
bull L1 NEG
Embedding NEG into L1
Conjecture (Explicit by Goemans Linial abt rsquo95)
Every NEG metric embeds into L1 with
O(1) (constant) distortion
Whatrsquos the connection to sparsity
Cuts and L1 Metrics
pS non-negative real for every subset S of [n]
d(ij) = pSδS(ij)
Fact d is isometrically embeddable in L1
Further Every L1 embeddable metric on n-points can be written as non-negative linear sum of cut-metrics on 1hellipn
Cut-Metrics on 12hellipn
δS(i j) = 1 if i j are separated by (SSc) = 0 otherwise
S Sc
Sparsest Cut asymp Optimizing Over L1
[Aumann Rabani 98 Linial London Rabinovich rsquo94]
bull LP-relaxation over all METRICSbull [Bourgain rsquo85] Every n-point metric embeds
into L1 with O(log n) distortionbull O(log n) factor approximation for Sparsity
i~j δS(i j)
---------- iltj δS(i j)
Minimize S V
Minimize d is L1
i~j d(i j)
---------- iiltj d(i j)
=
Metric Embeddings amp Sparsity
bull Optimizing over cuts asymp Optimizing over L1 metrics
bull SDP solution asymp Optimizing over NEG
bull Goemans-LinialARV Conjecture NEG embeds in L1 with O(1) distortion Integrality Gap is O(1)
bull Implies O(1) approx algo for estimating sparsity
Integrality Gap Lower Bound
Sparsest Cut Balanced Separator
log log n integrality gap instance
Khot-Vishnoi rsquo05 Krauthgamer-Rabani lsquo06Devanur-Khot-Saket-Vishnoi lsquo06
bull Disproves the GLARV Conjecture bull Previous best lower bound 116 [Zatloukal rsquo04]
Outline of this Talk Part I
1 Graph PartitioningMotivation amp HistorySDP Approach
2 Embedding Negative Type Metrics into L1 Metric Spaces and EmbeddabilityCut Cone asymp L1
Negative Type Metrics as SDP SolutionsLLRAR Connection Part II
1 Integrality Gap instance Hypercube and Cuts
bull Kahn-Kalai-Linial (isoperimetry of hypercube)The GraphThe SDP Solution
2 Conclusion
Outline of this Talk Part I
1 Graph PartitioningMotivation amp HistorySDP Approach
2 Embedding Negative Type Metrics into L1 Metric Spaces and EmbeddabilityCut Cone asymp L1
Negative Type Metrics as SDP SolutionsLLRAR Connection Part II
1 Integrality Gap instance Hypercube and Cuts
bull Kahn-Kalai-Linial (isoperimetry of hypercube)The GraphThe SDP Solution
2 Conclusion
Recall Integrality Gap Lower Bound
Sparsest Cut Balanced Separator
log log n integrality gap instance
Khot-Vishnoi rsquo05 Krauthgamer-Rabani rsquo06 Devanur-Khot-Saket-Vishnoi lsquo06
bull Disproves the GLARV Conjecture bull Previous best lower bound 116 [Zatloukal rsquo04]
Integrality Gap Lower Bound
Thm Construct graph G(1hellipnE) and unit vector assignment i -gt vi є Rn st
1 G is an ldquoexpanderrdquo every frac14 balanced cut has Ω(|E|(log log n)log n) edges (Kahn-Kalai-Linial)
2 ldquoLowrdquo SDP solution O(|E|log n)
3 Well-Separatedness Σiltj ||vi-vj||2 = n2
4 Triangle inequality d(ij)=||vi-vj||2 is a metric
Integrality gap Ω(log log n)
Starting Point Hypercube()
H=-11k
n = 2k
(111)
(-111)
(11-1)
(1-11)
(1-1-1)
(-1-11)
(-11-1) (-1-1-1)
Hypercube hellip H=-11k Advantages
bull Understand cuts in H tools from Fourier Analysis
bull Vertex is a vector in Rk starting point for SDP solution
But hellip
bull Hypercube has ldquosmallrdquo balanced cuts coordinate cuts have 1k fraction of the edges
Cuts in Hypercube Coordinate of edges = |E(H)|k
Edges across pairs of vertices differing in i-th bit
Cuts in Hypercube hellip
decompose into coordinate cuts
any balanced cut has a coord cut which contributes E(H)k2 edges
Kahn-Kalai-Linial
any balanced cut has a coordinate cut which contributes E(H) (log
k)k2 edges
Increasing Size of Balanced Cuts
consider balanced cuts in which coordinates are indistinguishable
(wrt to their contribution to the cut)
can be achieved by symmetrizing the hypercube
each coordinate contributes equally total E(H) (log k)k edges
eg 4-dim hypercube
(1111)
(-1-1-1-1)
(111-1) (-1111) (1-111) (11-11)
(-1-1-11) (1-1-1-1) (-11-1-1) (-1-11-1)
(11-1-1) (-111-1) (-1-111) (1-1-11)
(1-11-1) (-11-11)
More Formally hellipH=-11k with a rotation group acting on its coordinates
Partitions H into equivalence classes V1hellipVn
Each Vi is a vertex Edges are hypercube edges
G(VE) |E(G)|=|E(H)| k ~ log n
Balanced cuts in G correspond to balanced cuts in H
KKL any balanced cut (in H) has ldquoardquo coordinate cut which contributes E(H) (log k)k2 edges to the cut
Group is transitive ldquoeveryrdquo coordinate cut has the same contribution
Any balanced cut in G has ge E(G) (log k)k = E(G) (log log n)(log n)
Integrality Gap Lower Bound
Thm Construct graph G(1hellipnE) and unit vector assignment i -gt vi є Rn st
1 G is an ldquoexpanderrdquo every frac14 balanced cut has Ω(|E|(log log n)log n) edges (Kahn-Kalai-Linial)
2 ldquoLowrdquo SDP solution O(|E|log n)
3 Well-Separatedness Σiltj ||vi-vj||2 = n2
4 Triangle inequality d(ij)=||vi-vj||2 is a metric
Integrality gap Ω(log log n)
SDP Solution
(1111)
(-1-1-1-1)
(111-1) (-1111) (1-111) (11-11)
(-1-1-11) (1-1-1-1) (-11-1-1) (-1-11-1)
(11-1-1) (-111-1) (-1-111) (1-1-11)
(1-11-1) (-11-11)
Formally SDP SolutionVertex equivalence class x1 x2 hellip xk (rotations)
Vector (1radick) Σj xj
Observations
bull Edge across two nodes differing in one bit
Contribution to sdp ~ 1k sdp le |E(G)|k = |E(G)|(log n)
Triangle Inequality (little bit of work and case analysis)
For most classes x1 hellip xk is ldquonearly orthogonalrdquo
Hidden Gram-Schmidt Orthogonalization Tensoring Well-separatedness
Conclusion
bull (Simple) log log n integrality gap for SCBS
bull Close the gap between log log n and log n
bull [Lee-Naor rsquo06 Cheeger-Kleiner rsquo06] Another counter-example May give (log n)c
Thank you
- Slide 1
- CS Story Sparsity
- Applications
- Estimating Sparsity
- History
- Outline of this Talk
- Slide 7
- Quadratic Program for BS
- SDP for Balanced Separator
- Why is this a Relaxation
- Quadratic Program for BS hellip
- SDP for Balanced Separator hellip
- Geometry of Triangle Ineq
- Integrality Gap Upper Bound
- Slide 15
- Math Story Metric Embeddings
- Negative Type Metrics (squared-L2)
- Embedding NEG into L1
- Cuts and L1 Metrics
- Sparsest Cut asymp Optimizing Over L1
- Metric Embeddings amp Sparsity
- Integrality Gap Lower Bound
- Slide 23
- Slide 24
- Recall Integrality Gap Lower Bound
- Slide 26
- Starting Point Hypercube()
- Hypercube hellip
- Cuts in Hypercube Coordinate
- Cuts in Hypercube hellip
- Kahn-Kalai-Linial
- Increasing Size of Balanced Cuts
- eg 4-dim hypercube
- More Formally hellip
- Slide 35
- SDP Solution
- Formally SDP Solution
- Conclusion
-
Quadratic Program for BS hellip
i vi -11
Minimize frac14 |vi - vj |2
ij ε E
|vi - vj |2 = n2
iltj
Triangle Inequality i j k |vi - vj |2
+ |vj - vk |2 |vi - vk|2 (redundant)
Quadratic Program
Input G(VE)
Output (SSc)
st
|S||Sc| = n2
which minimizes
|E(SSc)|
Balanced Separator
SDP for Balanced Separator hellip
i vi -11
Minimize frac14 |vi - vj |2
ij ε E
|vi - vj |2 = n2
iltj
Triangle Inequality i j k |vi - vj |2+|vj - vk |2|vi - vk|2
Quadratic Program SDP Relaxation
i vi Rn ||vi||=1
Minimize frac14 || vi - vj ||2
ij E
Well-Separatedness
|| vi - vj ||2 = n2
iltj
Triangle Inequality i j k ||vi - vj||2 +||vj - vk||2 ||vi - vk||2
Still a relaxation hellip
Geometry of Triangle Ineq
le 90ovi
vk
vj
each step of length 1
t-steps length at-most radict
Rules out the embedding obtained by the spectral method
Integrality Gap Upper Bound
Arora-Rao-Vazirani rsquo04
O(radiclog n) for Sparsest Cut Balanced Separator
sdp within a factor of O(radiclog n) of the opt
Integrality gap max over all graphs on n vertices the ratio of
optsdp (as a function of n)
ARV conjectured that the integrality gap is upper bounded by some constant (independent of n)
Lack of any counterexample
Outline of this Talk Part I
1 Graph PartitioningMotivation amp HistorySDP Approach
2 Embedding Negative Type Metrics into L1 Metric Spaces and EmbeddabilityCut Cone asymp L1
Negative Type Metrics as SDP SolutionsLLRAR Connection
Part II 1 Integrality Gap instance
Hypercube and Cutsbull Kahn-Kalai-Linial (isoperimetry of hypercube)
The GraphThe SDP Solution
2 Conclusion
Math Story Metric Embeddings
bull Metric is a distance function d on [n] x [n] st d(i j) + d(j k) d(i k)
(triangle inequality)
bull Metric d embeds into metric with distortion 1 if there is a map φ st
ij d(i j) (φ(i) φ(j)) d(i j) (distances are preserved upto a factor of )
Negative Type Metrics (squared-L2)
bull d on 12hellipn is of negative type if there are vectors v1 v2 hellip vn
bull Such that d(i j) = || vi - vj ||2 satisfies triangle inequality
bull Same as i j k || vi - vj ||2
+ || vj - vk ||2 || vi - v k||2
bull NEG = class of such metrics bull arise as SDP solutions
bull L1 NEG
Embedding NEG into L1
Conjecture (Explicit by Goemans Linial abt rsquo95)
Every NEG metric embeds into L1 with
O(1) (constant) distortion
Whatrsquos the connection to sparsity
Cuts and L1 Metrics
pS non-negative real for every subset S of [n]
d(ij) = pSδS(ij)
Fact d is isometrically embeddable in L1
Further Every L1 embeddable metric on n-points can be written as non-negative linear sum of cut-metrics on 1hellipn
Cut-Metrics on 12hellipn
δS(i j) = 1 if i j are separated by (SSc) = 0 otherwise
S Sc
Sparsest Cut asymp Optimizing Over L1
[Aumann Rabani 98 Linial London Rabinovich rsquo94]
bull LP-relaxation over all METRICSbull [Bourgain rsquo85] Every n-point metric embeds
into L1 with O(log n) distortionbull O(log n) factor approximation for Sparsity
i~j δS(i j)
---------- iltj δS(i j)
Minimize S V
Minimize d is L1
i~j d(i j)
---------- iiltj d(i j)
=
Metric Embeddings amp Sparsity
bull Optimizing over cuts asymp Optimizing over L1 metrics
bull SDP solution asymp Optimizing over NEG
bull Goemans-LinialARV Conjecture NEG embeds in L1 with O(1) distortion Integrality Gap is O(1)
bull Implies O(1) approx algo for estimating sparsity
Integrality Gap Lower Bound
Sparsest Cut Balanced Separator
log log n integrality gap instance
Khot-Vishnoi rsquo05 Krauthgamer-Rabani lsquo06Devanur-Khot-Saket-Vishnoi lsquo06
bull Disproves the GLARV Conjecture bull Previous best lower bound 116 [Zatloukal rsquo04]
Outline of this Talk Part I
1 Graph PartitioningMotivation amp HistorySDP Approach
2 Embedding Negative Type Metrics into L1 Metric Spaces and EmbeddabilityCut Cone asymp L1
Negative Type Metrics as SDP SolutionsLLRAR Connection Part II
1 Integrality Gap instance Hypercube and Cuts
bull Kahn-Kalai-Linial (isoperimetry of hypercube)The GraphThe SDP Solution
2 Conclusion
Outline of this Talk Part I
1 Graph PartitioningMotivation amp HistorySDP Approach
2 Embedding Negative Type Metrics into L1 Metric Spaces and EmbeddabilityCut Cone asymp L1
Negative Type Metrics as SDP SolutionsLLRAR Connection Part II
1 Integrality Gap instance Hypercube and Cuts
bull Kahn-Kalai-Linial (isoperimetry of hypercube)The GraphThe SDP Solution
2 Conclusion
Recall Integrality Gap Lower Bound
Sparsest Cut Balanced Separator
log log n integrality gap instance
Khot-Vishnoi rsquo05 Krauthgamer-Rabani rsquo06 Devanur-Khot-Saket-Vishnoi lsquo06
bull Disproves the GLARV Conjecture bull Previous best lower bound 116 [Zatloukal rsquo04]
Integrality Gap Lower Bound
Thm Construct graph G(1hellipnE) and unit vector assignment i -gt vi є Rn st
1 G is an ldquoexpanderrdquo every frac14 balanced cut has Ω(|E|(log log n)log n) edges (Kahn-Kalai-Linial)
2 ldquoLowrdquo SDP solution O(|E|log n)
3 Well-Separatedness Σiltj ||vi-vj||2 = n2
4 Triangle inequality d(ij)=||vi-vj||2 is a metric
Integrality gap Ω(log log n)
Starting Point Hypercube()
H=-11k
n = 2k
(111)
(-111)
(11-1)
(1-11)
(1-1-1)
(-1-11)
(-11-1) (-1-1-1)
Hypercube hellip H=-11k Advantages
bull Understand cuts in H tools from Fourier Analysis
bull Vertex is a vector in Rk starting point for SDP solution
But hellip
bull Hypercube has ldquosmallrdquo balanced cuts coordinate cuts have 1k fraction of the edges
Cuts in Hypercube Coordinate of edges = |E(H)|k
Edges across pairs of vertices differing in i-th bit
Cuts in Hypercube hellip
decompose into coordinate cuts
any balanced cut has a coord cut which contributes E(H)k2 edges
Kahn-Kalai-Linial
any balanced cut has a coordinate cut which contributes E(H) (log
k)k2 edges
Increasing Size of Balanced Cuts
consider balanced cuts in which coordinates are indistinguishable
(wrt to their contribution to the cut)
can be achieved by symmetrizing the hypercube
each coordinate contributes equally total E(H) (log k)k edges
eg 4-dim hypercube
(1111)
(-1-1-1-1)
(111-1) (-1111) (1-111) (11-11)
(-1-1-11) (1-1-1-1) (-11-1-1) (-1-11-1)
(11-1-1) (-111-1) (-1-111) (1-1-11)
(1-11-1) (-11-11)
More Formally hellipH=-11k with a rotation group acting on its coordinates
Partitions H into equivalence classes V1hellipVn
Each Vi is a vertex Edges are hypercube edges
G(VE) |E(G)|=|E(H)| k ~ log n
Balanced cuts in G correspond to balanced cuts in H
KKL any balanced cut (in H) has ldquoardquo coordinate cut which contributes E(H) (log k)k2 edges to the cut
Group is transitive ldquoeveryrdquo coordinate cut has the same contribution
Any balanced cut in G has ge E(G) (log k)k = E(G) (log log n)(log n)
Integrality Gap Lower Bound
Thm Construct graph G(1hellipnE) and unit vector assignment i -gt vi є Rn st
1 G is an ldquoexpanderrdquo every frac14 balanced cut has Ω(|E|(log log n)log n) edges (Kahn-Kalai-Linial)
2 ldquoLowrdquo SDP solution O(|E|log n)
3 Well-Separatedness Σiltj ||vi-vj||2 = n2
4 Triangle inequality d(ij)=||vi-vj||2 is a metric
Integrality gap Ω(log log n)
SDP Solution
(1111)
(-1-1-1-1)
(111-1) (-1111) (1-111) (11-11)
(-1-1-11) (1-1-1-1) (-11-1-1) (-1-11-1)
(11-1-1) (-111-1) (-1-111) (1-1-11)
(1-11-1) (-11-11)
Formally SDP SolutionVertex equivalence class x1 x2 hellip xk (rotations)
Vector (1radick) Σj xj
Observations
bull Edge across two nodes differing in one bit
Contribution to sdp ~ 1k sdp le |E(G)|k = |E(G)|(log n)
Triangle Inequality (little bit of work and case analysis)
For most classes x1 hellip xk is ldquonearly orthogonalrdquo
Hidden Gram-Schmidt Orthogonalization Tensoring Well-separatedness
Conclusion
bull (Simple) log log n integrality gap for SCBS
bull Close the gap between log log n and log n
bull [Lee-Naor rsquo06 Cheeger-Kleiner rsquo06] Another counter-example May give (log n)c
Thank you
- Slide 1
- CS Story Sparsity
- Applications
- Estimating Sparsity
- History
- Outline of this Talk
- Slide 7
- Quadratic Program for BS
- SDP for Balanced Separator
- Why is this a Relaxation
- Quadratic Program for BS hellip
- SDP for Balanced Separator hellip
- Geometry of Triangle Ineq
- Integrality Gap Upper Bound
- Slide 15
- Math Story Metric Embeddings
- Negative Type Metrics (squared-L2)
- Embedding NEG into L1
- Cuts and L1 Metrics
- Sparsest Cut asymp Optimizing Over L1
- Metric Embeddings amp Sparsity
- Integrality Gap Lower Bound
- Slide 23
- Slide 24
- Recall Integrality Gap Lower Bound
- Slide 26
- Starting Point Hypercube()
- Hypercube hellip
- Cuts in Hypercube Coordinate
- Cuts in Hypercube hellip
- Kahn-Kalai-Linial
- Increasing Size of Balanced Cuts
- eg 4-dim hypercube
- More Formally hellip
- Slide 35
- SDP Solution
- Formally SDP Solution
- Conclusion
-
SDP for Balanced Separator hellip
i vi -11
Minimize frac14 |vi - vj |2
ij ε E
|vi - vj |2 = n2
iltj
Triangle Inequality i j k |vi - vj |2+|vj - vk |2|vi - vk|2
Quadratic Program SDP Relaxation
i vi Rn ||vi||=1
Minimize frac14 || vi - vj ||2
ij E
Well-Separatedness
|| vi - vj ||2 = n2
iltj
Triangle Inequality i j k ||vi - vj||2 +||vj - vk||2 ||vi - vk||2
Still a relaxation hellip
Geometry of Triangle Ineq
le 90ovi
vk
vj
each step of length 1
t-steps length at-most radict
Rules out the embedding obtained by the spectral method
Integrality Gap Upper Bound
Arora-Rao-Vazirani rsquo04
O(radiclog n) for Sparsest Cut Balanced Separator
sdp within a factor of O(radiclog n) of the opt
Integrality gap max over all graphs on n vertices the ratio of
optsdp (as a function of n)
ARV conjectured that the integrality gap is upper bounded by some constant (independent of n)
Lack of any counterexample
Outline of this Talk Part I
1 Graph PartitioningMotivation amp HistorySDP Approach
2 Embedding Negative Type Metrics into L1 Metric Spaces and EmbeddabilityCut Cone asymp L1
Negative Type Metrics as SDP SolutionsLLRAR Connection
Part II 1 Integrality Gap instance
Hypercube and Cutsbull Kahn-Kalai-Linial (isoperimetry of hypercube)
The GraphThe SDP Solution
2 Conclusion
Math Story Metric Embeddings
bull Metric is a distance function d on [n] x [n] st d(i j) + d(j k) d(i k)
(triangle inequality)
bull Metric d embeds into metric with distortion 1 if there is a map φ st
ij d(i j) (φ(i) φ(j)) d(i j) (distances are preserved upto a factor of )
Negative Type Metrics (squared-L2)
bull d on 12hellipn is of negative type if there are vectors v1 v2 hellip vn
bull Such that d(i j) = || vi - vj ||2 satisfies triangle inequality
bull Same as i j k || vi - vj ||2
+ || vj - vk ||2 || vi - v k||2
bull NEG = class of such metrics bull arise as SDP solutions
bull L1 NEG
Embedding NEG into L1
Conjecture (Explicit by Goemans Linial abt rsquo95)
Every NEG metric embeds into L1 with
O(1) (constant) distortion
Whatrsquos the connection to sparsity
Cuts and L1 Metrics
pS non-negative real for every subset S of [n]
d(ij) = pSδS(ij)
Fact d is isometrically embeddable in L1
Further Every L1 embeddable metric on n-points can be written as non-negative linear sum of cut-metrics on 1hellipn
Cut-Metrics on 12hellipn
δS(i j) = 1 if i j are separated by (SSc) = 0 otherwise
S Sc
Sparsest Cut asymp Optimizing Over L1
[Aumann Rabani 98 Linial London Rabinovich rsquo94]
bull LP-relaxation over all METRICSbull [Bourgain rsquo85] Every n-point metric embeds
into L1 with O(log n) distortionbull O(log n) factor approximation for Sparsity
i~j δS(i j)
---------- iltj δS(i j)
Minimize S V
Minimize d is L1
i~j d(i j)
---------- iiltj d(i j)
=
Metric Embeddings amp Sparsity
bull Optimizing over cuts asymp Optimizing over L1 metrics
bull SDP solution asymp Optimizing over NEG
bull Goemans-LinialARV Conjecture NEG embeds in L1 with O(1) distortion Integrality Gap is O(1)
bull Implies O(1) approx algo for estimating sparsity
Integrality Gap Lower Bound
Sparsest Cut Balanced Separator
log log n integrality gap instance
Khot-Vishnoi rsquo05 Krauthgamer-Rabani lsquo06Devanur-Khot-Saket-Vishnoi lsquo06
bull Disproves the GLARV Conjecture bull Previous best lower bound 116 [Zatloukal rsquo04]
Outline of this Talk Part I
1 Graph PartitioningMotivation amp HistorySDP Approach
2 Embedding Negative Type Metrics into L1 Metric Spaces and EmbeddabilityCut Cone asymp L1
Negative Type Metrics as SDP SolutionsLLRAR Connection Part II
1 Integrality Gap instance Hypercube and Cuts
bull Kahn-Kalai-Linial (isoperimetry of hypercube)The GraphThe SDP Solution
2 Conclusion
Outline of this Talk Part I
1 Graph PartitioningMotivation amp HistorySDP Approach
2 Embedding Negative Type Metrics into L1 Metric Spaces and EmbeddabilityCut Cone asymp L1
Negative Type Metrics as SDP SolutionsLLRAR Connection Part II
1 Integrality Gap instance Hypercube and Cuts
bull Kahn-Kalai-Linial (isoperimetry of hypercube)The GraphThe SDP Solution
2 Conclusion
Recall Integrality Gap Lower Bound
Sparsest Cut Balanced Separator
log log n integrality gap instance
Khot-Vishnoi rsquo05 Krauthgamer-Rabani rsquo06 Devanur-Khot-Saket-Vishnoi lsquo06
bull Disproves the GLARV Conjecture bull Previous best lower bound 116 [Zatloukal rsquo04]
Integrality Gap Lower Bound
Thm Construct graph G(1hellipnE) and unit vector assignment i -gt vi є Rn st
1 G is an ldquoexpanderrdquo every frac14 balanced cut has Ω(|E|(log log n)log n) edges (Kahn-Kalai-Linial)
2 ldquoLowrdquo SDP solution O(|E|log n)
3 Well-Separatedness Σiltj ||vi-vj||2 = n2
4 Triangle inequality d(ij)=||vi-vj||2 is a metric
Integrality gap Ω(log log n)
Starting Point Hypercube()
H=-11k
n = 2k
(111)
(-111)
(11-1)
(1-11)
(1-1-1)
(-1-11)
(-11-1) (-1-1-1)
Hypercube hellip H=-11k Advantages
bull Understand cuts in H tools from Fourier Analysis
bull Vertex is a vector in Rk starting point for SDP solution
But hellip
bull Hypercube has ldquosmallrdquo balanced cuts coordinate cuts have 1k fraction of the edges
Cuts in Hypercube Coordinate of edges = |E(H)|k
Edges across pairs of vertices differing in i-th bit
Cuts in Hypercube hellip
decompose into coordinate cuts
any balanced cut has a coord cut which contributes E(H)k2 edges
Kahn-Kalai-Linial
any balanced cut has a coordinate cut which contributes E(H) (log
k)k2 edges
Increasing Size of Balanced Cuts
consider balanced cuts in which coordinates are indistinguishable
(wrt to their contribution to the cut)
can be achieved by symmetrizing the hypercube
each coordinate contributes equally total E(H) (log k)k edges
eg 4-dim hypercube
(1111)
(-1-1-1-1)
(111-1) (-1111) (1-111) (11-11)
(-1-1-11) (1-1-1-1) (-11-1-1) (-1-11-1)
(11-1-1) (-111-1) (-1-111) (1-1-11)
(1-11-1) (-11-11)
More Formally hellipH=-11k with a rotation group acting on its coordinates
Partitions H into equivalence classes V1hellipVn
Each Vi is a vertex Edges are hypercube edges
G(VE) |E(G)|=|E(H)| k ~ log n
Balanced cuts in G correspond to balanced cuts in H
KKL any balanced cut (in H) has ldquoardquo coordinate cut which contributes E(H) (log k)k2 edges to the cut
Group is transitive ldquoeveryrdquo coordinate cut has the same contribution
Any balanced cut in G has ge E(G) (log k)k = E(G) (log log n)(log n)
Integrality Gap Lower Bound
Thm Construct graph G(1hellipnE) and unit vector assignment i -gt vi є Rn st
1 G is an ldquoexpanderrdquo every frac14 balanced cut has Ω(|E|(log log n)log n) edges (Kahn-Kalai-Linial)
2 ldquoLowrdquo SDP solution O(|E|log n)
3 Well-Separatedness Σiltj ||vi-vj||2 = n2
4 Triangle inequality d(ij)=||vi-vj||2 is a metric
Integrality gap Ω(log log n)
SDP Solution
(1111)
(-1-1-1-1)
(111-1) (-1111) (1-111) (11-11)
(-1-1-11) (1-1-1-1) (-11-1-1) (-1-11-1)
(11-1-1) (-111-1) (-1-111) (1-1-11)
(1-11-1) (-11-11)
Formally SDP SolutionVertex equivalence class x1 x2 hellip xk (rotations)
Vector (1radick) Σj xj
Observations
bull Edge across two nodes differing in one bit
Contribution to sdp ~ 1k sdp le |E(G)|k = |E(G)|(log n)
Triangle Inequality (little bit of work and case analysis)
For most classes x1 hellip xk is ldquonearly orthogonalrdquo
Hidden Gram-Schmidt Orthogonalization Tensoring Well-separatedness
Conclusion
bull (Simple) log log n integrality gap for SCBS
bull Close the gap between log log n and log n
bull [Lee-Naor rsquo06 Cheeger-Kleiner rsquo06] Another counter-example May give (log n)c
Thank you
- Slide 1
- CS Story Sparsity
- Applications
- Estimating Sparsity
- History
- Outline of this Talk
- Slide 7
- Quadratic Program for BS
- SDP for Balanced Separator
- Why is this a Relaxation
- Quadratic Program for BS hellip
- SDP for Balanced Separator hellip
- Geometry of Triangle Ineq
- Integrality Gap Upper Bound
- Slide 15
- Math Story Metric Embeddings
- Negative Type Metrics (squared-L2)
- Embedding NEG into L1
- Cuts and L1 Metrics
- Sparsest Cut asymp Optimizing Over L1
- Metric Embeddings amp Sparsity
- Integrality Gap Lower Bound
- Slide 23
- Slide 24
- Recall Integrality Gap Lower Bound
- Slide 26
- Starting Point Hypercube()
- Hypercube hellip
- Cuts in Hypercube Coordinate
- Cuts in Hypercube hellip
- Kahn-Kalai-Linial
- Increasing Size of Balanced Cuts
- eg 4-dim hypercube
- More Formally hellip
- Slide 35
- SDP Solution
- Formally SDP Solution
- Conclusion
-
Geometry of Triangle Ineq
le 90ovi
vk
vj
each step of length 1
t-steps length at-most radict
Rules out the embedding obtained by the spectral method
Integrality Gap Upper Bound
Arora-Rao-Vazirani rsquo04
O(radiclog n) for Sparsest Cut Balanced Separator
sdp within a factor of O(radiclog n) of the opt
Integrality gap max over all graphs on n vertices the ratio of
optsdp (as a function of n)
ARV conjectured that the integrality gap is upper bounded by some constant (independent of n)
Lack of any counterexample
Outline of this Talk Part I
1 Graph PartitioningMotivation amp HistorySDP Approach
2 Embedding Negative Type Metrics into L1 Metric Spaces and EmbeddabilityCut Cone asymp L1
Negative Type Metrics as SDP SolutionsLLRAR Connection
Part II 1 Integrality Gap instance
Hypercube and Cutsbull Kahn-Kalai-Linial (isoperimetry of hypercube)
The GraphThe SDP Solution
2 Conclusion
Math Story Metric Embeddings
bull Metric is a distance function d on [n] x [n] st d(i j) + d(j k) d(i k)
(triangle inequality)
bull Metric d embeds into metric with distortion 1 if there is a map φ st
ij d(i j) (φ(i) φ(j)) d(i j) (distances are preserved upto a factor of )
Negative Type Metrics (squared-L2)
bull d on 12hellipn is of negative type if there are vectors v1 v2 hellip vn
bull Such that d(i j) = || vi - vj ||2 satisfies triangle inequality
bull Same as i j k || vi - vj ||2
+ || vj - vk ||2 || vi - v k||2
bull NEG = class of such metrics bull arise as SDP solutions
bull L1 NEG
Embedding NEG into L1
Conjecture (Explicit by Goemans Linial abt rsquo95)
Every NEG metric embeds into L1 with
O(1) (constant) distortion
Whatrsquos the connection to sparsity
Cuts and L1 Metrics
pS non-negative real for every subset S of [n]
d(ij) = pSδS(ij)
Fact d is isometrically embeddable in L1
Further Every L1 embeddable metric on n-points can be written as non-negative linear sum of cut-metrics on 1hellipn
Cut-Metrics on 12hellipn
δS(i j) = 1 if i j are separated by (SSc) = 0 otherwise
S Sc
Sparsest Cut asymp Optimizing Over L1
[Aumann Rabani 98 Linial London Rabinovich rsquo94]
bull LP-relaxation over all METRICSbull [Bourgain rsquo85] Every n-point metric embeds
into L1 with O(log n) distortionbull O(log n) factor approximation for Sparsity
i~j δS(i j)
---------- iltj δS(i j)
Minimize S V
Minimize d is L1
i~j d(i j)
---------- iiltj d(i j)
=
Metric Embeddings amp Sparsity
bull Optimizing over cuts asymp Optimizing over L1 metrics
bull SDP solution asymp Optimizing over NEG
bull Goemans-LinialARV Conjecture NEG embeds in L1 with O(1) distortion Integrality Gap is O(1)
bull Implies O(1) approx algo for estimating sparsity
Integrality Gap Lower Bound
Sparsest Cut Balanced Separator
log log n integrality gap instance
Khot-Vishnoi rsquo05 Krauthgamer-Rabani lsquo06Devanur-Khot-Saket-Vishnoi lsquo06
bull Disproves the GLARV Conjecture bull Previous best lower bound 116 [Zatloukal rsquo04]
Outline of this Talk Part I
1 Graph PartitioningMotivation amp HistorySDP Approach
2 Embedding Negative Type Metrics into L1 Metric Spaces and EmbeddabilityCut Cone asymp L1
Negative Type Metrics as SDP SolutionsLLRAR Connection Part II
1 Integrality Gap instance Hypercube and Cuts
bull Kahn-Kalai-Linial (isoperimetry of hypercube)The GraphThe SDP Solution
2 Conclusion
Outline of this Talk Part I
1 Graph PartitioningMotivation amp HistorySDP Approach
2 Embedding Negative Type Metrics into L1 Metric Spaces and EmbeddabilityCut Cone asymp L1
Negative Type Metrics as SDP SolutionsLLRAR Connection Part II
1 Integrality Gap instance Hypercube and Cuts
bull Kahn-Kalai-Linial (isoperimetry of hypercube)The GraphThe SDP Solution
2 Conclusion
Recall Integrality Gap Lower Bound
Sparsest Cut Balanced Separator
log log n integrality gap instance
Khot-Vishnoi rsquo05 Krauthgamer-Rabani rsquo06 Devanur-Khot-Saket-Vishnoi lsquo06
bull Disproves the GLARV Conjecture bull Previous best lower bound 116 [Zatloukal rsquo04]
Integrality Gap Lower Bound
Thm Construct graph G(1hellipnE) and unit vector assignment i -gt vi є Rn st
1 G is an ldquoexpanderrdquo every frac14 balanced cut has Ω(|E|(log log n)log n) edges (Kahn-Kalai-Linial)
2 ldquoLowrdquo SDP solution O(|E|log n)
3 Well-Separatedness Σiltj ||vi-vj||2 = n2
4 Triangle inequality d(ij)=||vi-vj||2 is a metric
Integrality gap Ω(log log n)
Starting Point Hypercube()
H=-11k
n = 2k
(111)
(-111)
(11-1)
(1-11)
(1-1-1)
(-1-11)
(-11-1) (-1-1-1)
Hypercube hellip H=-11k Advantages
bull Understand cuts in H tools from Fourier Analysis
bull Vertex is a vector in Rk starting point for SDP solution
But hellip
bull Hypercube has ldquosmallrdquo balanced cuts coordinate cuts have 1k fraction of the edges
Cuts in Hypercube Coordinate of edges = |E(H)|k
Edges across pairs of vertices differing in i-th bit
Cuts in Hypercube hellip
decompose into coordinate cuts
any balanced cut has a coord cut which contributes E(H)k2 edges
Kahn-Kalai-Linial
any balanced cut has a coordinate cut which contributes E(H) (log
k)k2 edges
Increasing Size of Balanced Cuts
consider balanced cuts in which coordinates are indistinguishable
(wrt to their contribution to the cut)
can be achieved by symmetrizing the hypercube
each coordinate contributes equally total E(H) (log k)k edges
eg 4-dim hypercube
(1111)
(-1-1-1-1)
(111-1) (-1111) (1-111) (11-11)
(-1-1-11) (1-1-1-1) (-11-1-1) (-1-11-1)
(11-1-1) (-111-1) (-1-111) (1-1-11)
(1-11-1) (-11-11)
More Formally hellipH=-11k with a rotation group acting on its coordinates
Partitions H into equivalence classes V1hellipVn
Each Vi is a vertex Edges are hypercube edges
G(VE) |E(G)|=|E(H)| k ~ log n
Balanced cuts in G correspond to balanced cuts in H
KKL any balanced cut (in H) has ldquoardquo coordinate cut which contributes E(H) (log k)k2 edges to the cut
Group is transitive ldquoeveryrdquo coordinate cut has the same contribution
Any balanced cut in G has ge E(G) (log k)k = E(G) (log log n)(log n)
Integrality Gap Lower Bound
Thm Construct graph G(1hellipnE) and unit vector assignment i -gt vi є Rn st
1 G is an ldquoexpanderrdquo every frac14 balanced cut has Ω(|E|(log log n)log n) edges (Kahn-Kalai-Linial)
2 ldquoLowrdquo SDP solution O(|E|log n)
3 Well-Separatedness Σiltj ||vi-vj||2 = n2
4 Triangle inequality d(ij)=||vi-vj||2 is a metric
Integrality gap Ω(log log n)
SDP Solution
(1111)
(-1-1-1-1)
(111-1) (-1111) (1-111) (11-11)
(-1-1-11) (1-1-1-1) (-11-1-1) (-1-11-1)
(11-1-1) (-111-1) (-1-111) (1-1-11)
(1-11-1) (-11-11)
Formally SDP SolutionVertex equivalence class x1 x2 hellip xk (rotations)
Vector (1radick) Σj xj
Observations
bull Edge across two nodes differing in one bit
Contribution to sdp ~ 1k sdp le |E(G)|k = |E(G)|(log n)
Triangle Inequality (little bit of work and case analysis)
For most classes x1 hellip xk is ldquonearly orthogonalrdquo
Hidden Gram-Schmidt Orthogonalization Tensoring Well-separatedness
Conclusion
bull (Simple) log log n integrality gap for SCBS
bull Close the gap between log log n and log n
bull [Lee-Naor rsquo06 Cheeger-Kleiner rsquo06] Another counter-example May give (log n)c
Thank you
- Slide 1
- CS Story Sparsity
- Applications
- Estimating Sparsity
- History
- Outline of this Talk
- Slide 7
- Quadratic Program for BS
- SDP for Balanced Separator
- Why is this a Relaxation
- Quadratic Program for BS hellip
- SDP for Balanced Separator hellip
- Geometry of Triangle Ineq
- Integrality Gap Upper Bound
- Slide 15
- Math Story Metric Embeddings
- Negative Type Metrics (squared-L2)
- Embedding NEG into L1
- Cuts and L1 Metrics
- Sparsest Cut asymp Optimizing Over L1
- Metric Embeddings amp Sparsity
- Integrality Gap Lower Bound
- Slide 23
- Slide 24
- Recall Integrality Gap Lower Bound
- Slide 26
- Starting Point Hypercube()
- Hypercube hellip
- Cuts in Hypercube Coordinate
- Cuts in Hypercube hellip
- Kahn-Kalai-Linial
- Increasing Size of Balanced Cuts
- eg 4-dim hypercube
- More Formally hellip
- Slide 35
- SDP Solution
- Formally SDP Solution
- Conclusion
-
Integrality Gap Upper Bound
Arora-Rao-Vazirani rsquo04
O(radiclog n) for Sparsest Cut Balanced Separator
sdp within a factor of O(radiclog n) of the opt
Integrality gap max over all graphs on n vertices the ratio of
optsdp (as a function of n)
ARV conjectured that the integrality gap is upper bounded by some constant (independent of n)
Lack of any counterexample
Outline of this Talk Part I
1 Graph PartitioningMotivation amp HistorySDP Approach
2 Embedding Negative Type Metrics into L1 Metric Spaces and EmbeddabilityCut Cone asymp L1
Negative Type Metrics as SDP SolutionsLLRAR Connection
Part II 1 Integrality Gap instance
Hypercube and Cutsbull Kahn-Kalai-Linial (isoperimetry of hypercube)
The GraphThe SDP Solution
2 Conclusion
Math Story Metric Embeddings
bull Metric is a distance function d on [n] x [n] st d(i j) + d(j k) d(i k)
(triangle inequality)
bull Metric d embeds into metric with distortion 1 if there is a map φ st
ij d(i j) (φ(i) φ(j)) d(i j) (distances are preserved upto a factor of )
Negative Type Metrics (squared-L2)
bull d on 12hellipn is of negative type if there are vectors v1 v2 hellip vn
bull Such that d(i j) = || vi - vj ||2 satisfies triangle inequality
bull Same as i j k || vi - vj ||2
+ || vj - vk ||2 || vi - v k||2
bull NEG = class of such metrics bull arise as SDP solutions
bull L1 NEG
Embedding NEG into L1
Conjecture (Explicit by Goemans Linial abt rsquo95)
Every NEG metric embeds into L1 with
O(1) (constant) distortion
Whatrsquos the connection to sparsity
Cuts and L1 Metrics
pS non-negative real for every subset S of [n]
d(ij) = pSδS(ij)
Fact d is isometrically embeddable in L1
Further Every L1 embeddable metric on n-points can be written as non-negative linear sum of cut-metrics on 1hellipn
Cut-Metrics on 12hellipn
δS(i j) = 1 if i j are separated by (SSc) = 0 otherwise
S Sc
Sparsest Cut asymp Optimizing Over L1
[Aumann Rabani 98 Linial London Rabinovich rsquo94]
bull LP-relaxation over all METRICSbull [Bourgain rsquo85] Every n-point metric embeds
into L1 with O(log n) distortionbull O(log n) factor approximation for Sparsity
i~j δS(i j)
---------- iltj δS(i j)
Minimize S V
Minimize d is L1
i~j d(i j)
---------- iiltj d(i j)
=
Metric Embeddings amp Sparsity
bull Optimizing over cuts asymp Optimizing over L1 metrics
bull SDP solution asymp Optimizing over NEG
bull Goemans-LinialARV Conjecture NEG embeds in L1 with O(1) distortion Integrality Gap is O(1)
bull Implies O(1) approx algo for estimating sparsity
Integrality Gap Lower Bound
Sparsest Cut Balanced Separator
log log n integrality gap instance
Khot-Vishnoi rsquo05 Krauthgamer-Rabani lsquo06Devanur-Khot-Saket-Vishnoi lsquo06
bull Disproves the GLARV Conjecture bull Previous best lower bound 116 [Zatloukal rsquo04]
Outline of this Talk Part I
1 Graph PartitioningMotivation amp HistorySDP Approach
2 Embedding Negative Type Metrics into L1 Metric Spaces and EmbeddabilityCut Cone asymp L1
Negative Type Metrics as SDP SolutionsLLRAR Connection Part II
1 Integrality Gap instance Hypercube and Cuts
bull Kahn-Kalai-Linial (isoperimetry of hypercube)The GraphThe SDP Solution
2 Conclusion
Outline of this Talk Part I
1 Graph PartitioningMotivation amp HistorySDP Approach
2 Embedding Negative Type Metrics into L1 Metric Spaces and EmbeddabilityCut Cone asymp L1
Negative Type Metrics as SDP SolutionsLLRAR Connection Part II
1 Integrality Gap instance Hypercube and Cuts
bull Kahn-Kalai-Linial (isoperimetry of hypercube)The GraphThe SDP Solution
2 Conclusion
Recall Integrality Gap Lower Bound
Sparsest Cut Balanced Separator
log log n integrality gap instance
Khot-Vishnoi rsquo05 Krauthgamer-Rabani rsquo06 Devanur-Khot-Saket-Vishnoi lsquo06
bull Disproves the GLARV Conjecture bull Previous best lower bound 116 [Zatloukal rsquo04]
Integrality Gap Lower Bound
Thm Construct graph G(1hellipnE) and unit vector assignment i -gt vi є Rn st
1 G is an ldquoexpanderrdquo every frac14 balanced cut has Ω(|E|(log log n)log n) edges (Kahn-Kalai-Linial)
2 ldquoLowrdquo SDP solution O(|E|log n)
3 Well-Separatedness Σiltj ||vi-vj||2 = n2
4 Triangle inequality d(ij)=||vi-vj||2 is a metric
Integrality gap Ω(log log n)
Starting Point Hypercube()
H=-11k
n = 2k
(111)
(-111)
(11-1)
(1-11)
(1-1-1)
(-1-11)
(-11-1) (-1-1-1)
Hypercube hellip H=-11k Advantages
bull Understand cuts in H tools from Fourier Analysis
bull Vertex is a vector in Rk starting point for SDP solution
But hellip
bull Hypercube has ldquosmallrdquo balanced cuts coordinate cuts have 1k fraction of the edges
Cuts in Hypercube Coordinate of edges = |E(H)|k
Edges across pairs of vertices differing in i-th bit
Cuts in Hypercube hellip
decompose into coordinate cuts
any balanced cut has a coord cut which contributes E(H)k2 edges
Kahn-Kalai-Linial
any balanced cut has a coordinate cut which contributes E(H) (log
k)k2 edges
Increasing Size of Balanced Cuts
consider balanced cuts in which coordinates are indistinguishable
(wrt to their contribution to the cut)
can be achieved by symmetrizing the hypercube
each coordinate contributes equally total E(H) (log k)k edges
eg 4-dim hypercube
(1111)
(-1-1-1-1)
(111-1) (-1111) (1-111) (11-11)
(-1-1-11) (1-1-1-1) (-11-1-1) (-1-11-1)
(11-1-1) (-111-1) (-1-111) (1-1-11)
(1-11-1) (-11-11)
More Formally hellipH=-11k with a rotation group acting on its coordinates
Partitions H into equivalence classes V1hellipVn
Each Vi is a vertex Edges are hypercube edges
G(VE) |E(G)|=|E(H)| k ~ log n
Balanced cuts in G correspond to balanced cuts in H
KKL any balanced cut (in H) has ldquoardquo coordinate cut which contributes E(H) (log k)k2 edges to the cut
Group is transitive ldquoeveryrdquo coordinate cut has the same contribution
Any balanced cut in G has ge E(G) (log k)k = E(G) (log log n)(log n)
Integrality Gap Lower Bound
Thm Construct graph G(1hellipnE) and unit vector assignment i -gt vi є Rn st
1 G is an ldquoexpanderrdquo every frac14 balanced cut has Ω(|E|(log log n)log n) edges (Kahn-Kalai-Linial)
2 ldquoLowrdquo SDP solution O(|E|log n)
3 Well-Separatedness Σiltj ||vi-vj||2 = n2
4 Triangle inequality d(ij)=||vi-vj||2 is a metric
Integrality gap Ω(log log n)
SDP Solution
(1111)
(-1-1-1-1)
(111-1) (-1111) (1-111) (11-11)
(-1-1-11) (1-1-1-1) (-11-1-1) (-1-11-1)
(11-1-1) (-111-1) (-1-111) (1-1-11)
(1-11-1) (-11-11)
Formally SDP SolutionVertex equivalence class x1 x2 hellip xk (rotations)
Vector (1radick) Σj xj
Observations
bull Edge across two nodes differing in one bit
Contribution to sdp ~ 1k sdp le |E(G)|k = |E(G)|(log n)
Triangle Inequality (little bit of work and case analysis)
For most classes x1 hellip xk is ldquonearly orthogonalrdquo
Hidden Gram-Schmidt Orthogonalization Tensoring Well-separatedness
Conclusion
bull (Simple) log log n integrality gap for SCBS
bull Close the gap between log log n and log n
bull [Lee-Naor rsquo06 Cheeger-Kleiner rsquo06] Another counter-example May give (log n)c
Thank you
- Slide 1
- CS Story Sparsity
- Applications
- Estimating Sparsity
- History
- Outline of this Talk
- Slide 7
- Quadratic Program for BS
- SDP for Balanced Separator
- Why is this a Relaxation
- Quadratic Program for BS hellip
- SDP for Balanced Separator hellip
- Geometry of Triangle Ineq
- Integrality Gap Upper Bound
- Slide 15
- Math Story Metric Embeddings
- Negative Type Metrics (squared-L2)
- Embedding NEG into L1
- Cuts and L1 Metrics
- Sparsest Cut asymp Optimizing Over L1
- Metric Embeddings amp Sparsity
- Integrality Gap Lower Bound
- Slide 23
- Slide 24
- Recall Integrality Gap Lower Bound
- Slide 26
- Starting Point Hypercube()
- Hypercube hellip
- Cuts in Hypercube Coordinate
- Cuts in Hypercube hellip
- Kahn-Kalai-Linial
- Increasing Size of Balanced Cuts
- eg 4-dim hypercube
- More Formally hellip
- Slide 35
- SDP Solution
- Formally SDP Solution
- Conclusion
-
Outline of this Talk Part I
1 Graph PartitioningMotivation amp HistorySDP Approach
2 Embedding Negative Type Metrics into L1 Metric Spaces and EmbeddabilityCut Cone asymp L1
Negative Type Metrics as SDP SolutionsLLRAR Connection
Part II 1 Integrality Gap instance
Hypercube and Cutsbull Kahn-Kalai-Linial (isoperimetry of hypercube)
The GraphThe SDP Solution
2 Conclusion
Math Story Metric Embeddings
bull Metric is a distance function d on [n] x [n] st d(i j) + d(j k) d(i k)
(triangle inequality)
bull Metric d embeds into metric with distortion 1 if there is a map φ st
ij d(i j) (φ(i) φ(j)) d(i j) (distances are preserved upto a factor of )
Negative Type Metrics (squared-L2)
bull d on 12hellipn is of negative type if there are vectors v1 v2 hellip vn
bull Such that d(i j) = || vi - vj ||2 satisfies triangle inequality
bull Same as i j k || vi - vj ||2
+ || vj - vk ||2 || vi - v k||2
bull NEG = class of such metrics bull arise as SDP solutions
bull L1 NEG
Embedding NEG into L1
Conjecture (Explicit by Goemans Linial abt rsquo95)
Every NEG metric embeds into L1 with
O(1) (constant) distortion
Whatrsquos the connection to sparsity
Cuts and L1 Metrics
pS non-negative real for every subset S of [n]
d(ij) = pSδS(ij)
Fact d is isometrically embeddable in L1
Further Every L1 embeddable metric on n-points can be written as non-negative linear sum of cut-metrics on 1hellipn
Cut-Metrics on 12hellipn
δS(i j) = 1 if i j are separated by (SSc) = 0 otherwise
S Sc
Sparsest Cut asymp Optimizing Over L1
[Aumann Rabani 98 Linial London Rabinovich rsquo94]
bull LP-relaxation over all METRICSbull [Bourgain rsquo85] Every n-point metric embeds
into L1 with O(log n) distortionbull O(log n) factor approximation for Sparsity
i~j δS(i j)
---------- iltj δS(i j)
Minimize S V
Minimize d is L1
i~j d(i j)
---------- iiltj d(i j)
=
Metric Embeddings amp Sparsity
bull Optimizing over cuts asymp Optimizing over L1 metrics
bull SDP solution asymp Optimizing over NEG
bull Goemans-LinialARV Conjecture NEG embeds in L1 with O(1) distortion Integrality Gap is O(1)
bull Implies O(1) approx algo for estimating sparsity
Integrality Gap Lower Bound
Sparsest Cut Balanced Separator
log log n integrality gap instance
Khot-Vishnoi rsquo05 Krauthgamer-Rabani lsquo06Devanur-Khot-Saket-Vishnoi lsquo06
bull Disproves the GLARV Conjecture bull Previous best lower bound 116 [Zatloukal rsquo04]
Outline of this Talk Part I
1 Graph PartitioningMotivation amp HistorySDP Approach
2 Embedding Negative Type Metrics into L1 Metric Spaces and EmbeddabilityCut Cone asymp L1
Negative Type Metrics as SDP SolutionsLLRAR Connection Part II
1 Integrality Gap instance Hypercube and Cuts
bull Kahn-Kalai-Linial (isoperimetry of hypercube)The GraphThe SDP Solution
2 Conclusion
Outline of this Talk Part I
1 Graph PartitioningMotivation amp HistorySDP Approach
2 Embedding Negative Type Metrics into L1 Metric Spaces and EmbeddabilityCut Cone asymp L1
Negative Type Metrics as SDP SolutionsLLRAR Connection Part II
1 Integrality Gap instance Hypercube and Cuts
bull Kahn-Kalai-Linial (isoperimetry of hypercube)The GraphThe SDP Solution
2 Conclusion
Recall Integrality Gap Lower Bound
Sparsest Cut Balanced Separator
log log n integrality gap instance
Khot-Vishnoi rsquo05 Krauthgamer-Rabani rsquo06 Devanur-Khot-Saket-Vishnoi lsquo06
bull Disproves the GLARV Conjecture bull Previous best lower bound 116 [Zatloukal rsquo04]
Integrality Gap Lower Bound
Thm Construct graph G(1hellipnE) and unit vector assignment i -gt vi є Rn st
1 G is an ldquoexpanderrdquo every frac14 balanced cut has Ω(|E|(log log n)log n) edges (Kahn-Kalai-Linial)
2 ldquoLowrdquo SDP solution O(|E|log n)
3 Well-Separatedness Σiltj ||vi-vj||2 = n2
4 Triangle inequality d(ij)=||vi-vj||2 is a metric
Integrality gap Ω(log log n)
Starting Point Hypercube()
H=-11k
n = 2k
(111)
(-111)
(11-1)
(1-11)
(1-1-1)
(-1-11)
(-11-1) (-1-1-1)
Hypercube hellip H=-11k Advantages
bull Understand cuts in H tools from Fourier Analysis
bull Vertex is a vector in Rk starting point for SDP solution
But hellip
bull Hypercube has ldquosmallrdquo balanced cuts coordinate cuts have 1k fraction of the edges
Cuts in Hypercube Coordinate of edges = |E(H)|k
Edges across pairs of vertices differing in i-th bit
Cuts in Hypercube hellip
decompose into coordinate cuts
any balanced cut has a coord cut which contributes E(H)k2 edges
Kahn-Kalai-Linial
any balanced cut has a coordinate cut which contributes E(H) (log
k)k2 edges
Increasing Size of Balanced Cuts
consider balanced cuts in which coordinates are indistinguishable
(wrt to their contribution to the cut)
can be achieved by symmetrizing the hypercube
each coordinate contributes equally total E(H) (log k)k edges
eg 4-dim hypercube
(1111)
(-1-1-1-1)
(111-1) (-1111) (1-111) (11-11)
(-1-1-11) (1-1-1-1) (-11-1-1) (-1-11-1)
(11-1-1) (-111-1) (-1-111) (1-1-11)
(1-11-1) (-11-11)
More Formally hellipH=-11k with a rotation group acting on its coordinates
Partitions H into equivalence classes V1hellipVn
Each Vi is a vertex Edges are hypercube edges
G(VE) |E(G)|=|E(H)| k ~ log n
Balanced cuts in G correspond to balanced cuts in H
KKL any balanced cut (in H) has ldquoardquo coordinate cut which contributes E(H) (log k)k2 edges to the cut
Group is transitive ldquoeveryrdquo coordinate cut has the same contribution
Any balanced cut in G has ge E(G) (log k)k = E(G) (log log n)(log n)
Integrality Gap Lower Bound
Thm Construct graph G(1hellipnE) and unit vector assignment i -gt vi є Rn st
1 G is an ldquoexpanderrdquo every frac14 balanced cut has Ω(|E|(log log n)log n) edges (Kahn-Kalai-Linial)
2 ldquoLowrdquo SDP solution O(|E|log n)
3 Well-Separatedness Σiltj ||vi-vj||2 = n2
4 Triangle inequality d(ij)=||vi-vj||2 is a metric
Integrality gap Ω(log log n)
SDP Solution
(1111)
(-1-1-1-1)
(111-1) (-1111) (1-111) (11-11)
(-1-1-11) (1-1-1-1) (-11-1-1) (-1-11-1)
(11-1-1) (-111-1) (-1-111) (1-1-11)
(1-11-1) (-11-11)
Formally SDP SolutionVertex equivalence class x1 x2 hellip xk (rotations)
Vector (1radick) Σj xj
Observations
bull Edge across two nodes differing in one bit
Contribution to sdp ~ 1k sdp le |E(G)|k = |E(G)|(log n)
Triangle Inequality (little bit of work and case analysis)
For most classes x1 hellip xk is ldquonearly orthogonalrdquo
Hidden Gram-Schmidt Orthogonalization Tensoring Well-separatedness
Conclusion
bull (Simple) log log n integrality gap for SCBS
bull Close the gap between log log n and log n
bull [Lee-Naor rsquo06 Cheeger-Kleiner rsquo06] Another counter-example May give (log n)c
Thank you
- Slide 1
- CS Story Sparsity
- Applications
- Estimating Sparsity
- History
- Outline of this Talk
- Slide 7
- Quadratic Program for BS
- SDP for Balanced Separator
- Why is this a Relaxation
- Quadratic Program for BS hellip
- SDP for Balanced Separator hellip
- Geometry of Triangle Ineq
- Integrality Gap Upper Bound
- Slide 15
- Math Story Metric Embeddings
- Negative Type Metrics (squared-L2)
- Embedding NEG into L1
- Cuts and L1 Metrics
- Sparsest Cut asymp Optimizing Over L1
- Metric Embeddings amp Sparsity
- Integrality Gap Lower Bound
- Slide 23
- Slide 24
- Recall Integrality Gap Lower Bound
- Slide 26
- Starting Point Hypercube()
- Hypercube hellip
- Cuts in Hypercube Coordinate
- Cuts in Hypercube hellip
- Kahn-Kalai-Linial
- Increasing Size of Balanced Cuts
- eg 4-dim hypercube
- More Formally hellip
- Slide 35
- SDP Solution
- Formally SDP Solution
- Conclusion
-
Math Story Metric Embeddings
bull Metric is a distance function d on [n] x [n] st d(i j) + d(j k) d(i k)
(triangle inequality)
bull Metric d embeds into metric with distortion 1 if there is a map φ st
ij d(i j) (φ(i) φ(j)) d(i j) (distances are preserved upto a factor of )
Negative Type Metrics (squared-L2)
bull d on 12hellipn is of negative type if there are vectors v1 v2 hellip vn
bull Such that d(i j) = || vi - vj ||2 satisfies triangle inequality
bull Same as i j k || vi - vj ||2
+ || vj - vk ||2 || vi - v k||2
bull NEG = class of such metrics bull arise as SDP solutions
bull L1 NEG
Embedding NEG into L1
Conjecture (Explicit by Goemans Linial abt rsquo95)
Every NEG metric embeds into L1 with
O(1) (constant) distortion
Whatrsquos the connection to sparsity
Cuts and L1 Metrics
pS non-negative real for every subset S of [n]
d(ij) = pSδS(ij)
Fact d is isometrically embeddable in L1
Further Every L1 embeddable metric on n-points can be written as non-negative linear sum of cut-metrics on 1hellipn
Cut-Metrics on 12hellipn
δS(i j) = 1 if i j are separated by (SSc) = 0 otherwise
S Sc
Sparsest Cut asymp Optimizing Over L1
[Aumann Rabani 98 Linial London Rabinovich rsquo94]
bull LP-relaxation over all METRICSbull [Bourgain rsquo85] Every n-point metric embeds
into L1 with O(log n) distortionbull O(log n) factor approximation for Sparsity
i~j δS(i j)
---------- iltj δS(i j)
Minimize S V
Minimize d is L1
i~j d(i j)
---------- iiltj d(i j)
=
Metric Embeddings amp Sparsity
bull Optimizing over cuts asymp Optimizing over L1 metrics
bull SDP solution asymp Optimizing over NEG
bull Goemans-LinialARV Conjecture NEG embeds in L1 with O(1) distortion Integrality Gap is O(1)
bull Implies O(1) approx algo for estimating sparsity
Integrality Gap Lower Bound
Sparsest Cut Balanced Separator
log log n integrality gap instance
Khot-Vishnoi rsquo05 Krauthgamer-Rabani lsquo06Devanur-Khot-Saket-Vishnoi lsquo06
bull Disproves the GLARV Conjecture bull Previous best lower bound 116 [Zatloukal rsquo04]
Outline of this Talk Part I
1 Graph PartitioningMotivation amp HistorySDP Approach
2 Embedding Negative Type Metrics into L1 Metric Spaces and EmbeddabilityCut Cone asymp L1
Negative Type Metrics as SDP SolutionsLLRAR Connection Part II
1 Integrality Gap instance Hypercube and Cuts
bull Kahn-Kalai-Linial (isoperimetry of hypercube)The GraphThe SDP Solution
2 Conclusion
Outline of this Talk Part I
1 Graph PartitioningMotivation amp HistorySDP Approach
2 Embedding Negative Type Metrics into L1 Metric Spaces and EmbeddabilityCut Cone asymp L1
Negative Type Metrics as SDP SolutionsLLRAR Connection Part II
1 Integrality Gap instance Hypercube and Cuts
bull Kahn-Kalai-Linial (isoperimetry of hypercube)The GraphThe SDP Solution
2 Conclusion
Recall Integrality Gap Lower Bound
Sparsest Cut Balanced Separator
log log n integrality gap instance
Khot-Vishnoi rsquo05 Krauthgamer-Rabani rsquo06 Devanur-Khot-Saket-Vishnoi lsquo06
bull Disproves the GLARV Conjecture bull Previous best lower bound 116 [Zatloukal rsquo04]
Integrality Gap Lower Bound
Thm Construct graph G(1hellipnE) and unit vector assignment i -gt vi є Rn st
1 G is an ldquoexpanderrdquo every frac14 balanced cut has Ω(|E|(log log n)log n) edges (Kahn-Kalai-Linial)
2 ldquoLowrdquo SDP solution O(|E|log n)
3 Well-Separatedness Σiltj ||vi-vj||2 = n2
4 Triangle inequality d(ij)=||vi-vj||2 is a metric
Integrality gap Ω(log log n)
Starting Point Hypercube()
H=-11k
n = 2k
(111)
(-111)
(11-1)
(1-11)
(1-1-1)
(-1-11)
(-11-1) (-1-1-1)
Hypercube hellip H=-11k Advantages
bull Understand cuts in H tools from Fourier Analysis
bull Vertex is a vector in Rk starting point for SDP solution
But hellip
bull Hypercube has ldquosmallrdquo balanced cuts coordinate cuts have 1k fraction of the edges
Cuts in Hypercube Coordinate of edges = |E(H)|k
Edges across pairs of vertices differing in i-th bit
Cuts in Hypercube hellip
decompose into coordinate cuts
any balanced cut has a coord cut which contributes E(H)k2 edges
Kahn-Kalai-Linial
any balanced cut has a coordinate cut which contributes E(H) (log
k)k2 edges
Increasing Size of Balanced Cuts
consider balanced cuts in which coordinates are indistinguishable
(wrt to their contribution to the cut)
can be achieved by symmetrizing the hypercube
each coordinate contributes equally total E(H) (log k)k edges
eg 4-dim hypercube
(1111)
(-1-1-1-1)
(111-1) (-1111) (1-111) (11-11)
(-1-1-11) (1-1-1-1) (-11-1-1) (-1-11-1)
(11-1-1) (-111-1) (-1-111) (1-1-11)
(1-11-1) (-11-11)
More Formally hellipH=-11k with a rotation group acting on its coordinates
Partitions H into equivalence classes V1hellipVn
Each Vi is a vertex Edges are hypercube edges
G(VE) |E(G)|=|E(H)| k ~ log n
Balanced cuts in G correspond to balanced cuts in H
KKL any balanced cut (in H) has ldquoardquo coordinate cut which contributes E(H) (log k)k2 edges to the cut
Group is transitive ldquoeveryrdquo coordinate cut has the same contribution
Any balanced cut in G has ge E(G) (log k)k = E(G) (log log n)(log n)
Integrality Gap Lower Bound
Thm Construct graph G(1hellipnE) and unit vector assignment i -gt vi є Rn st
1 G is an ldquoexpanderrdquo every frac14 balanced cut has Ω(|E|(log log n)log n) edges (Kahn-Kalai-Linial)
2 ldquoLowrdquo SDP solution O(|E|log n)
3 Well-Separatedness Σiltj ||vi-vj||2 = n2
4 Triangle inequality d(ij)=||vi-vj||2 is a metric
Integrality gap Ω(log log n)
SDP Solution
(1111)
(-1-1-1-1)
(111-1) (-1111) (1-111) (11-11)
(-1-1-11) (1-1-1-1) (-11-1-1) (-1-11-1)
(11-1-1) (-111-1) (-1-111) (1-1-11)
(1-11-1) (-11-11)
Formally SDP SolutionVertex equivalence class x1 x2 hellip xk (rotations)
Vector (1radick) Σj xj
Observations
bull Edge across two nodes differing in one bit
Contribution to sdp ~ 1k sdp le |E(G)|k = |E(G)|(log n)
Triangle Inequality (little bit of work and case analysis)
For most classes x1 hellip xk is ldquonearly orthogonalrdquo
Hidden Gram-Schmidt Orthogonalization Tensoring Well-separatedness
Conclusion
bull (Simple) log log n integrality gap for SCBS
bull Close the gap between log log n and log n
bull [Lee-Naor rsquo06 Cheeger-Kleiner rsquo06] Another counter-example May give (log n)c
Thank you
- Slide 1
- CS Story Sparsity
- Applications
- Estimating Sparsity
- History
- Outline of this Talk
- Slide 7
- Quadratic Program for BS
- SDP for Balanced Separator
- Why is this a Relaxation
- Quadratic Program for BS hellip
- SDP for Balanced Separator hellip
- Geometry of Triangle Ineq
- Integrality Gap Upper Bound
- Slide 15
- Math Story Metric Embeddings
- Negative Type Metrics (squared-L2)
- Embedding NEG into L1
- Cuts and L1 Metrics
- Sparsest Cut asymp Optimizing Over L1
- Metric Embeddings amp Sparsity
- Integrality Gap Lower Bound
- Slide 23
- Slide 24
- Recall Integrality Gap Lower Bound
- Slide 26
- Starting Point Hypercube()
- Hypercube hellip
- Cuts in Hypercube Coordinate
- Cuts in Hypercube hellip
- Kahn-Kalai-Linial
- Increasing Size of Balanced Cuts
- eg 4-dim hypercube
- More Formally hellip
- Slide 35
- SDP Solution
- Formally SDP Solution
- Conclusion
-
Negative Type Metrics (squared-L2)
bull d on 12hellipn is of negative type if there are vectors v1 v2 hellip vn
bull Such that d(i j) = || vi - vj ||2 satisfies triangle inequality
bull Same as i j k || vi - vj ||2
+ || vj - vk ||2 || vi - v k||2
bull NEG = class of such metrics bull arise as SDP solutions
bull L1 NEG
Embedding NEG into L1
Conjecture (Explicit by Goemans Linial abt rsquo95)
Every NEG metric embeds into L1 with
O(1) (constant) distortion
Whatrsquos the connection to sparsity
Cuts and L1 Metrics
pS non-negative real for every subset S of [n]
d(ij) = pSδS(ij)
Fact d is isometrically embeddable in L1
Further Every L1 embeddable metric on n-points can be written as non-negative linear sum of cut-metrics on 1hellipn
Cut-Metrics on 12hellipn
δS(i j) = 1 if i j are separated by (SSc) = 0 otherwise
S Sc
Sparsest Cut asymp Optimizing Over L1
[Aumann Rabani 98 Linial London Rabinovich rsquo94]
bull LP-relaxation over all METRICSbull [Bourgain rsquo85] Every n-point metric embeds
into L1 with O(log n) distortionbull O(log n) factor approximation for Sparsity
i~j δS(i j)
---------- iltj δS(i j)
Minimize S V
Minimize d is L1
i~j d(i j)
---------- iiltj d(i j)
=
Metric Embeddings amp Sparsity
bull Optimizing over cuts asymp Optimizing over L1 metrics
bull SDP solution asymp Optimizing over NEG
bull Goemans-LinialARV Conjecture NEG embeds in L1 with O(1) distortion Integrality Gap is O(1)
bull Implies O(1) approx algo for estimating sparsity
Integrality Gap Lower Bound
Sparsest Cut Balanced Separator
log log n integrality gap instance
Khot-Vishnoi rsquo05 Krauthgamer-Rabani lsquo06Devanur-Khot-Saket-Vishnoi lsquo06
bull Disproves the GLARV Conjecture bull Previous best lower bound 116 [Zatloukal rsquo04]
Outline of this Talk Part I
1 Graph PartitioningMotivation amp HistorySDP Approach
2 Embedding Negative Type Metrics into L1 Metric Spaces and EmbeddabilityCut Cone asymp L1
Negative Type Metrics as SDP SolutionsLLRAR Connection Part II
1 Integrality Gap instance Hypercube and Cuts
bull Kahn-Kalai-Linial (isoperimetry of hypercube)The GraphThe SDP Solution
2 Conclusion
Outline of this Talk Part I
1 Graph PartitioningMotivation amp HistorySDP Approach
2 Embedding Negative Type Metrics into L1 Metric Spaces and EmbeddabilityCut Cone asymp L1
Negative Type Metrics as SDP SolutionsLLRAR Connection Part II
1 Integrality Gap instance Hypercube and Cuts
bull Kahn-Kalai-Linial (isoperimetry of hypercube)The GraphThe SDP Solution
2 Conclusion
Recall Integrality Gap Lower Bound
Sparsest Cut Balanced Separator
log log n integrality gap instance
Khot-Vishnoi rsquo05 Krauthgamer-Rabani rsquo06 Devanur-Khot-Saket-Vishnoi lsquo06
bull Disproves the GLARV Conjecture bull Previous best lower bound 116 [Zatloukal rsquo04]
Integrality Gap Lower Bound
Thm Construct graph G(1hellipnE) and unit vector assignment i -gt vi є Rn st
1 G is an ldquoexpanderrdquo every frac14 balanced cut has Ω(|E|(log log n)log n) edges (Kahn-Kalai-Linial)
2 ldquoLowrdquo SDP solution O(|E|log n)
3 Well-Separatedness Σiltj ||vi-vj||2 = n2
4 Triangle inequality d(ij)=||vi-vj||2 is a metric
Integrality gap Ω(log log n)
Starting Point Hypercube()
H=-11k
n = 2k
(111)
(-111)
(11-1)
(1-11)
(1-1-1)
(-1-11)
(-11-1) (-1-1-1)
Hypercube hellip H=-11k Advantages
bull Understand cuts in H tools from Fourier Analysis
bull Vertex is a vector in Rk starting point for SDP solution
But hellip
bull Hypercube has ldquosmallrdquo balanced cuts coordinate cuts have 1k fraction of the edges
Cuts in Hypercube Coordinate of edges = |E(H)|k
Edges across pairs of vertices differing in i-th bit
Cuts in Hypercube hellip
decompose into coordinate cuts
any balanced cut has a coord cut which contributes E(H)k2 edges
Kahn-Kalai-Linial
any balanced cut has a coordinate cut which contributes E(H) (log
k)k2 edges
Increasing Size of Balanced Cuts
consider balanced cuts in which coordinates are indistinguishable
(wrt to their contribution to the cut)
can be achieved by symmetrizing the hypercube
each coordinate contributes equally total E(H) (log k)k edges
eg 4-dim hypercube
(1111)
(-1-1-1-1)
(111-1) (-1111) (1-111) (11-11)
(-1-1-11) (1-1-1-1) (-11-1-1) (-1-11-1)
(11-1-1) (-111-1) (-1-111) (1-1-11)
(1-11-1) (-11-11)
More Formally hellipH=-11k with a rotation group acting on its coordinates
Partitions H into equivalence classes V1hellipVn
Each Vi is a vertex Edges are hypercube edges
G(VE) |E(G)|=|E(H)| k ~ log n
Balanced cuts in G correspond to balanced cuts in H
KKL any balanced cut (in H) has ldquoardquo coordinate cut which contributes E(H) (log k)k2 edges to the cut
Group is transitive ldquoeveryrdquo coordinate cut has the same contribution
Any balanced cut in G has ge E(G) (log k)k = E(G) (log log n)(log n)
Integrality Gap Lower Bound
Thm Construct graph G(1hellipnE) and unit vector assignment i -gt vi є Rn st
1 G is an ldquoexpanderrdquo every frac14 balanced cut has Ω(|E|(log log n)log n) edges (Kahn-Kalai-Linial)
2 ldquoLowrdquo SDP solution O(|E|log n)
3 Well-Separatedness Σiltj ||vi-vj||2 = n2
4 Triangle inequality d(ij)=||vi-vj||2 is a metric
Integrality gap Ω(log log n)
SDP Solution
(1111)
(-1-1-1-1)
(111-1) (-1111) (1-111) (11-11)
(-1-1-11) (1-1-1-1) (-11-1-1) (-1-11-1)
(11-1-1) (-111-1) (-1-111) (1-1-11)
(1-11-1) (-11-11)
Formally SDP SolutionVertex equivalence class x1 x2 hellip xk (rotations)
Vector (1radick) Σj xj
Observations
bull Edge across two nodes differing in one bit
Contribution to sdp ~ 1k sdp le |E(G)|k = |E(G)|(log n)
Triangle Inequality (little bit of work and case analysis)
For most classes x1 hellip xk is ldquonearly orthogonalrdquo
Hidden Gram-Schmidt Orthogonalization Tensoring Well-separatedness
Conclusion
bull (Simple) log log n integrality gap for SCBS
bull Close the gap between log log n and log n
bull [Lee-Naor rsquo06 Cheeger-Kleiner rsquo06] Another counter-example May give (log n)c
Thank you
- Slide 1
- CS Story Sparsity
- Applications
- Estimating Sparsity
- History
- Outline of this Talk
- Slide 7
- Quadratic Program for BS
- SDP for Balanced Separator
- Why is this a Relaxation
- Quadratic Program for BS hellip
- SDP for Balanced Separator hellip
- Geometry of Triangle Ineq
- Integrality Gap Upper Bound
- Slide 15
- Math Story Metric Embeddings
- Negative Type Metrics (squared-L2)
- Embedding NEG into L1
- Cuts and L1 Metrics
- Sparsest Cut asymp Optimizing Over L1
- Metric Embeddings amp Sparsity
- Integrality Gap Lower Bound
- Slide 23
- Slide 24
- Recall Integrality Gap Lower Bound
- Slide 26
- Starting Point Hypercube()
- Hypercube hellip
- Cuts in Hypercube Coordinate
- Cuts in Hypercube hellip
- Kahn-Kalai-Linial
- Increasing Size of Balanced Cuts
- eg 4-dim hypercube
- More Formally hellip
- Slide 35
- SDP Solution
- Formally SDP Solution
- Conclusion
-
Embedding NEG into L1
Conjecture (Explicit by Goemans Linial abt rsquo95)
Every NEG metric embeds into L1 with
O(1) (constant) distortion
Whatrsquos the connection to sparsity
Cuts and L1 Metrics
pS non-negative real for every subset S of [n]
d(ij) = pSδS(ij)
Fact d is isometrically embeddable in L1
Further Every L1 embeddable metric on n-points can be written as non-negative linear sum of cut-metrics on 1hellipn
Cut-Metrics on 12hellipn
δS(i j) = 1 if i j are separated by (SSc) = 0 otherwise
S Sc
Sparsest Cut asymp Optimizing Over L1
[Aumann Rabani 98 Linial London Rabinovich rsquo94]
bull LP-relaxation over all METRICSbull [Bourgain rsquo85] Every n-point metric embeds
into L1 with O(log n) distortionbull O(log n) factor approximation for Sparsity
i~j δS(i j)
---------- iltj δS(i j)
Minimize S V
Minimize d is L1
i~j d(i j)
---------- iiltj d(i j)
=
Metric Embeddings amp Sparsity
bull Optimizing over cuts asymp Optimizing over L1 metrics
bull SDP solution asymp Optimizing over NEG
bull Goemans-LinialARV Conjecture NEG embeds in L1 with O(1) distortion Integrality Gap is O(1)
bull Implies O(1) approx algo for estimating sparsity
Integrality Gap Lower Bound
Sparsest Cut Balanced Separator
log log n integrality gap instance
Khot-Vishnoi rsquo05 Krauthgamer-Rabani lsquo06Devanur-Khot-Saket-Vishnoi lsquo06
bull Disproves the GLARV Conjecture bull Previous best lower bound 116 [Zatloukal rsquo04]
Outline of this Talk Part I
1 Graph PartitioningMotivation amp HistorySDP Approach
2 Embedding Negative Type Metrics into L1 Metric Spaces and EmbeddabilityCut Cone asymp L1
Negative Type Metrics as SDP SolutionsLLRAR Connection Part II
1 Integrality Gap instance Hypercube and Cuts
bull Kahn-Kalai-Linial (isoperimetry of hypercube)The GraphThe SDP Solution
2 Conclusion
Outline of this Talk Part I
1 Graph PartitioningMotivation amp HistorySDP Approach
2 Embedding Negative Type Metrics into L1 Metric Spaces and EmbeddabilityCut Cone asymp L1
Negative Type Metrics as SDP SolutionsLLRAR Connection Part II
1 Integrality Gap instance Hypercube and Cuts
bull Kahn-Kalai-Linial (isoperimetry of hypercube)The GraphThe SDP Solution
2 Conclusion
Recall Integrality Gap Lower Bound
Sparsest Cut Balanced Separator
log log n integrality gap instance
Khot-Vishnoi rsquo05 Krauthgamer-Rabani rsquo06 Devanur-Khot-Saket-Vishnoi lsquo06
bull Disproves the GLARV Conjecture bull Previous best lower bound 116 [Zatloukal rsquo04]
Integrality Gap Lower Bound
Thm Construct graph G(1hellipnE) and unit vector assignment i -gt vi є Rn st
1 G is an ldquoexpanderrdquo every frac14 balanced cut has Ω(|E|(log log n)log n) edges (Kahn-Kalai-Linial)
2 ldquoLowrdquo SDP solution O(|E|log n)
3 Well-Separatedness Σiltj ||vi-vj||2 = n2
4 Triangle inequality d(ij)=||vi-vj||2 is a metric
Integrality gap Ω(log log n)
Starting Point Hypercube()
H=-11k
n = 2k
(111)
(-111)
(11-1)
(1-11)
(1-1-1)
(-1-11)
(-11-1) (-1-1-1)
Hypercube hellip H=-11k Advantages
bull Understand cuts in H tools from Fourier Analysis
bull Vertex is a vector in Rk starting point for SDP solution
But hellip
bull Hypercube has ldquosmallrdquo balanced cuts coordinate cuts have 1k fraction of the edges
Cuts in Hypercube Coordinate of edges = |E(H)|k
Edges across pairs of vertices differing in i-th bit
Cuts in Hypercube hellip
decompose into coordinate cuts
any balanced cut has a coord cut which contributes E(H)k2 edges
Kahn-Kalai-Linial
any balanced cut has a coordinate cut which contributes E(H) (log
k)k2 edges
Increasing Size of Balanced Cuts
consider balanced cuts in which coordinates are indistinguishable
(wrt to their contribution to the cut)
can be achieved by symmetrizing the hypercube
each coordinate contributes equally total E(H) (log k)k edges
eg 4-dim hypercube
(1111)
(-1-1-1-1)
(111-1) (-1111) (1-111) (11-11)
(-1-1-11) (1-1-1-1) (-11-1-1) (-1-11-1)
(11-1-1) (-111-1) (-1-111) (1-1-11)
(1-11-1) (-11-11)
More Formally hellipH=-11k with a rotation group acting on its coordinates
Partitions H into equivalence classes V1hellipVn
Each Vi is a vertex Edges are hypercube edges
G(VE) |E(G)|=|E(H)| k ~ log n
Balanced cuts in G correspond to balanced cuts in H
KKL any balanced cut (in H) has ldquoardquo coordinate cut which contributes E(H) (log k)k2 edges to the cut
Group is transitive ldquoeveryrdquo coordinate cut has the same contribution
Any balanced cut in G has ge E(G) (log k)k = E(G) (log log n)(log n)
Integrality Gap Lower Bound
Thm Construct graph G(1hellipnE) and unit vector assignment i -gt vi є Rn st
1 G is an ldquoexpanderrdquo every frac14 balanced cut has Ω(|E|(log log n)log n) edges (Kahn-Kalai-Linial)
2 ldquoLowrdquo SDP solution O(|E|log n)
3 Well-Separatedness Σiltj ||vi-vj||2 = n2
4 Triangle inequality d(ij)=||vi-vj||2 is a metric
Integrality gap Ω(log log n)
SDP Solution
(1111)
(-1-1-1-1)
(111-1) (-1111) (1-111) (11-11)
(-1-1-11) (1-1-1-1) (-11-1-1) (-1-11-1)
(11-1-1) (-111-1) (-1-111) (1-1-11)
(1-11-1) (-11-11)
Formally SDP SolutionVertex equivalence class x1 x2 hellip xk (rotations)
Vector (1radick) Σj xj
Observations
bull Edge across two nodes differing in one bit
Contribution to sdp ~ 1k sdp le |E(G)|k = |E(G)|(log n)
Triangle Inequality (little bit of work and case analysis)
For most classes x1 hellip xk is ldquonearly orthogonalrdquo
Hidden Gram-Schmidt Orthogonalization Tensoring Well-separatedness
Conclusion
bull (Simple) log log n integrality gap for SCBS
bull Close the gap between log log n and log n
bull [Lee-Naor rsquo06 Cheeger-Kleiner rsquo06] Another counter-example May give (log n)c
Thank you
- Slide 1
- CS Story Sparsity
- Applications
- Estimating Sparsity
- History
- Outline of this Talk
- Slide 7
- Quadratic Program for BS
- SDP for Balanced Separator
- Why is this a Relaxation
- Quadratic Program for BS hellip
- SDP for Balanced Separator hellip
- Geometry of Triangle Ineq
- Integrality Gap Upper Bound
- Slide 15
- Math Story Metric Embeddings
- Negative Type Metrics (squared-L2)
- Embedding NEG into L1
- Cuts and L1 Metrics
- Sparsest Cut asymp Optimizing Over L1
- Metric Embeddings amp Sparsity
- Integrality Gap Lower Bound
- Slide 23
- Slide 24
- Recall Integrality Gap Lower Bound
- Slide 26
- Starting Point Hypercube()
- Hypercube hellip
- Cuts in Hypercube Coordinate
- Cuts in Hypercube hellip
- Kahn-Kalai-Linial
- Increasing Size of Balanced Cuts
- eg 4-dim hypercube
- More Formally hellip
- Slide 35
- SDP Solution
- Formally SDP Solution
- Conclusion
-
Cuts and L1 Metrics
pS non-negative real for every subset S of [n]
d(ij) = pSδS(ij)
Fact d is isometrically embeddable in L1
Further Every L1 embeddable metric on n-points can be written as non-negative linear sum of cut-metrics on 1hellipn
Cut-Metrics on 12hellipn
δS(i j) = 1 if i j are separated by (SSc) = 0 otherwise
S Sc
Sparsest Cut asymp Optimizing Over L1
[Aumann Rabani 98 Linial London Rabinovich rsquo94]
bull LP-relaxation over all METRICSbull [Bourgain rsquo85] Every n-point metric embeds
into L1 with O(log n) distortionbull O(log n) factor approximation for Sparsity
i~j δS(i j)
---------- iltj δS(i j)
Minimize S V
Minimize d is L1
i~j d(i j)
---------- iiltj d(i j)
=
Metric Embeddings amp Sparsity
bull Optimizing over cuts asymp Optimizing over L1 metrics
bull SDP solution asymp Optimizing over NEG
bull Goemans-LinialARV Conjecture NEG embeds in L1 with O(1) distortion Integrality Gap is O(1)
bull Implies O(1) approx algo for estimating sparsity
Integrality Gap Lower Bound
Sparsest Cut Balanced Separator
log log n integrality gap instance
Khot-Vishnoi rsquo05 Krauthgamer-Rabani lsquo06Devanur-Khot-Saket-Vishnoi lsquo06
bull Disproves the GLARV Conjecture bull Previous best lower bound 116 [Zatloukal rsquo04]
Outline of this Talk Part I
1 Graph PartitioningMotivation amp HistorySDP Approach
2 Embedding Negative Type Metrics into L1 Metric Spaces and EmbeddabilityCut Cone asymp L1
Negative Type Metrics as SDP SolutionsLLRAR Connection Part II
1 Integrality Gap instance Hypercube and Cuts
bull Kahn-Kalai-Linial (isoperimetry of hypercube)The GraphThe SDP Solution
2 Conclusion
Outline of this Talk Part I
1 Graph PartitioningMotivation amp HistorySDP Approach
2 Embedding Negative Type Metrics into L1 Metric Spaces and EmbeddabilityCut Cone asymp L1
Negative Type Metrics as SDP SolutionsLLRAR Connection Part II
1 Integrality Gap instance Hypercube and Cuts
bull Kahn-Kalai-Linial (isoperimetry of hypercube)The GraphThe SDP Solution
2 Conclusion
Recall Integrality Gap Lower Bound
Sparsest Cut Balanced Separator
log log n integrality gap instance
Khot-Vishnoi rsquo05 Krauthgamer-Rabani rsquo06 Devanur-Khot-Saket-Vishnoi lsquo06
bull Disproves the GLARV Conjecture bull Previous best lower bound 116 [Zatloukal rsquo04]
Integrality Gap Lower Bound
Thm Construct graph G(1hellipnE) and unit vector assignment i -gt vi є Rn st
1 G is an ldquoexpanderrdquo every frac14 balanced cut has Ω(|E|(log log n)log n) edges (Kahn-Kalai-Linial)
2 ldquoLowrdquo SDP solution O(|E|log n)
3 Well-Separatedness Σiltj ||vi-vj||2 = n2
4 Triangle inequality d(ij)=||vi-vj||2 is a metric
Integrality gap Ω(log log n)
Starting Point Hypercube()
H=-11k
n = 2k
(111)
(-111)
(11-1)
(1-11)
(1-1-1)
(-1-11)
(-11-1) (-1-1-1)
Hypercube hellip H=-11k Advantages
bull Understand cuts in H tools from Fourier Analysis
bull Vertex is a vector in Rk starting point for SDP solution
But hellip
bull Hypercube has ldquosmallrdquo balanced cuts coordinate cuts have 1k fraction of the edges
Cuts in Hypercube Coordinate of edges = |E(H)|k
Edges across pairs of vertices differing in i-th bit
Cuts in Hypercube hellip
decompose into coordinate cuts
any balanced cut has a coord cut which contributes E(H)k2 edges
Kahn-Kalai-Linial
any balanced cut has a coordinate cut which contributes E(H) (log
k)k2 edges
Increasing Size of Balanced Cuts
consider balanced cuts in which coordinates are indistinguishable
(wrt to their contribution to the cut)
can be achieved by symmetrizing the hypercube
each coordinate contributes equally total E(H) (log k)k edges
eg 4-dim hypercube
(1111)
(-1-1-1-1)
(111-1) (-1111) (1-111) (11-11)
(-1-1-11) (1-1-1-1) (-11-1-1) (-1-11-1)
(11-1-1) (-111-1) (-1-111) (1-1-11)
(1-11-1) (-11-11)
More Formally hellipH=-11k with a rotation group acting on its coordinates
Partitions H into equivalence classes V1hellipVn
Each Vi is a vertex Edges are hypercube edges
G(VE) |E(G)|=|E(H)| k ~ log n
Balanced cuts in G correspond to balanced cuts in H
KKL any balanced cut (in H) has ldquoardquo coordinate cut which contributes E(H) (log k)k2 edges to the cut
Group is transitive ldquoeveryrdquo coordinate cut has the same contribution
Any balanced cut in G has ge E(G) (log k)k = E(G) (log log n)(log n)
Integrality Gap Lower Bound
Thm Construct graph G(1hellipnE) and unit vector assignment i -gt vi є Rn st
1 G is an ldquoexpanderrdquo every frac14 balanced cut has Ω(|E|(log log n)log n) edges (Kahn-Kalai-Linial)
2 ldquoLowrdquo SDP solution O(|E|log n)
3 Well-Separatedness Σiltj ||vi-vj||2 = n2
4 Triangle inequality d(ij)=||vi-vj||2 is a metric
Integrality gap Ω(log log n)
SDP Solution
(1111)
(-1-1-1-1)
(111-1) (-1111) (1-111) (11-11)
(-1-1-11) (1-1-1-1) (-11-1-1) (-1-11-1)
(11-1-1) (-111-1) (-1-111) (1-1-11)
(1-11-1) (-11-11)
Formally SDP SolutionVertex equivalence class x1 x2 hellip xk (rotations)
Vector (1radick) Σj xj
Observations
bull Edge across two nodes differing in one bit
Contribution to sdp ~ 1k sdp le |E(G)|k = |E(G)|(log n)
Triangle Inequality (little bit of work and case analysis)
For most classes x1 hellip xk is ldquonearly orthogonalrdquo
Hidden Gram-Schmidt Orthogonalization Tensoring Well-separatedness
Conclusion
bull (Simple) log log n integrality gap for SCBS
bull Close the gap between log log n and log n
bull [Lee-Naor rsquo06 Cheeger-Kleiner rsquo06] Another counter-example May give (log n)c
Thank you
- Slide 1
- CS Story Sparsity
- Applications
- Estimating Sparsity
- History
- Outline of this Talk
- Slide 7
- Quadratic Program for BS
- SDP for Balanced Separator
- Why is this a Relaxation
- Quadratic Program for BS hellip
- SDP for Balanced Separator hellip
- Geometry of Triangle Ineq
- Integrality Gap Upper Bound
- Slide 15
- Math Story Metric Embeddings
- Negative Type Metrics (squared-L2)
- Embedding NEG into L1
- Cuts and L1 Metrics
- Sparsest Cut asymp Optimizing Over L1
- Metric Embeddings amp Sparsity
- Integrality Gap Lower Bound
- Slide 23
- Slide 24
- Recall Integrality Gap Lower Bound
- Slide 26
- Starting Point Hypercube()
- Hypercube hellip
- Cuts in Hypercube Coordinate
- Cuts in Hypercube hellip
- Kahn-Kalai-Linial
- Increasing Size of Balanced Cuts
- eg 4-dim hypercube
- More Formally hellip
- Slide 35
- SDP Solution
- Formally SDP Solution
- Conclusion
-
Sparsest Cut asymp Optimizing Over L1
[Aumann Rabani 98 Linial London Rabinovich rsquo94]
bull LP-relaxation over all METRICSbull [Bourgain rsquo85] Every n-point metric embeds
into L1 with O(log n) distortionbull O(log n) factor approximation for Sparsity
i~j δS(i j)
---------- iltj δS(i j)
Minimize S V
Minimize d is L1
i~j d(i j)
---------- iiltj d(i j)
=
Metric Embeddings amp Sparsity
bull Optimizing over cuts asymp Optimizing over L1 metrics
bull SDP solution asymp Optimizing over NEG
bull Goemans-LinialARV Conjecture NEG embeds in L1 with O(1) distortion Integrality Gap is O(1)
bull Implies O(1) approx algo for estimating sparsity
Integrality Gap Lower Bound
Sparsest Cut Balanced Separator
log log n integrality gap instance
Khot-Vishnoi rsquo05 Krauthgamer-Rabani lsquo06Devanur-Khot-Saket-Vishnoi lsquo06
bull Disproves the GLARV Conjecture bull Previous best lower bound 116 [Zatloukal rsquo04]
Outline of this Talk Part I
1 Graph PartitioningMotivation amp HistorySDP Approach
2 Embedding Negative Type Metrics into L1 Metric Spaces and EmbeddabilityCut Cone asymp L1
Negative Type Metrics as SDP SolutionsLLRAR Connection Part II
1 Integrality Gap instance Hypercube and Cuts
bull Kahn-Kalai-Linial (isoperimetry of hypercube)The GraphThe SDP Solution
2 Conclusion
Outline of this Talk Part I
1 Graph PartitioningMotivation amp HistorySDP Approach
2 Embedding Negative Type Metrics into L1 Metric Spaces and EmbeddabilityCut Cone asymp L1
Negative Type Metrics as SDP SolutionsLLRAR Connection Part II
1 Integrality Gap instance Hypercube and Cuts
bull Kahn-Kalai-Linial (isoperimetry of hypercube)The GraphThe SDP Solution
2 Conclusion
Recall Integrality Gap Lower Bound
Sparsest Cut Balanced Separator
log log n integrality gap instance
Khot-Vishnoi rsquo05 Krauthgamer-Rabani rsquo06 Devanur-Khot-Saket-Vishnoi lsquo06
bull Disproves the GLARV Conjecture bull Previous best lower bound 116 [Zatloukal rsquo04]
Integrality Gap Lower Bound
Thm Construct graph G(1hellipnE) and unit vector assignment i -gt vi є Rn st
1 G is an ldquoexpanderrdquo every frac14 balanced cut has Ω(|E|(log log n)log n) edges (Kahn-Kalai-Linial)
2 ldquoLowrdquo SDP solution O(|E|log n)
3 Well-Separatedness Σiltj ||vi-vj||2 = n2
4 Triangle inequality d(ij)=||vi-vj||2 is a metric
Integrality gap Ω(log log n)
Starting Point Hypercube()
H=-11k
n = 2k
(111)
(-111)
(11-1)
(1-11)
(1-1-1)
(-1-11)
(-11-1) (-1-1-1)
Hypercube hellip H=-11k Advantages
bull Understand cuts in H tools from Fourier Analysis
bull Vertex is a vector in Rk starting point for SDP solution
But hellip
bull Hypercube has ldquosmallrdquo balanced cuts coordinate cuts have 1k fraction of the edges
Cuts in Hypercube Coordinate of edges = |E(H)|k
Edges across pairs of vertices differing in i-th bit
Cuts in Hypercube hellip
decompose into coordinate cuts
any balanced cut has a coord cut which contributes E(H)k2 edges
Kahn-Kalai-Linial
any balanced cut has a coordinate cut which contributes E(H) (log
k)k2 edges
Increasing Size of Balanced Cuts
consider balanced cuts in which coordinates are indistinguishable
(wrt to their contribution to the cut)
can be achieved by symmetrizing the hypercube
each coordinate contributes equally total E(H) (log k)k edges
eg 4-dim hypercube
(1111)
(-1-1-1-1)
(111-1) (-1111) (1-111) (11-11)
(-1-1-11) (1-1-1-1) (-11-1-1) (-1-11-1)
(11-1-1) (-111-1) (-1-111) (1-1-11)
(1-11-1) (-11-11)
More Formally hellipH=-11k with a rotation group acting on its coordinates
Partitions H into equivalence classes V1hellipVn
Each Vi is a vertex Edges are hypercube edges
G(VE) |E(G)|=|E(H)| k ~ log n
Balanced cuts in G correspond to balanced cuts in H
KKL any balanced cut (in H) has ldquoardquo coordinate cut which contributes E(H) (log k)k2 edges to the cut
Group is transitive ldquoeveryrdquo coordinate cut has the same contribution
Any balanced cut in G has ge E(G) (log k)k = E(G) (log log n)(log n)
Integrality Gap Lower Bound
Thm Construct graph G(1hellipnE) and unit vector assignment i -gt vi є Rn st
1 G is an ldquoexpanderrdquo every frac14 balanced cut has Ω(|E|(log log n)log n) edges (Kahn-Kalai-Linial)
2 ldquoLowrdquo SDP solution O(|E|log n)
3 Well-Separatedness Σiltj ||vi-vj||2 = n2
4 Triangle inequality d(ij)=||vi-vj||2 is a metric
Integrality gap Ω(log log n)
SDP Solution
(1111)
(-1-1-1-1)
(111-1) (-1111) (1-111) (11-11)
(-1-1-11) (1-1-1-1) (-11-1-1) (-1-11-1)
(11-1-1) (-111-1) (-1-111) (1-1-11)
(1-11-1) (-11-11)
Formally SDP SolutionVertex equivalence class x1 x2 hellip xk (rotations)
Vector (1radick) Σj xj
Observations
bull Edge across two nodes differing in one bit
Contribution to sdp ~ 1k sdp le |E(G)|k = |E(G)|(log n)
Triangle Inequality (little bit of work and case analysis)
For most classes x1 hellip xk is ldquonearly orthogonalrdquo
Hidden Gram-Schmidt Orthogonalization Tensoring Well-separatedness
Conclusion
bull (Simple) log log n integrality gap for SCBS
bull Close the gap between log log n and log n
bull [Lee-Naor rsquo06 Cheeger-Kleiner rsquo06] Another counter-example May give (log n)c
Thank you
- Slide 1
- CS Story Sparsity
- Applications
- Estimating Sparsity
- History
- Outline of this Talk
- Slide 7
- Quadratic Program for BS
- SDP for Balanced Separator
- Why is this a Relaxation
- Quadratic Program for BS hellip
- SDP for Balanced Separator hellip
- Geometry of Triangle Ineq
- Integrality Gap Upper Bound
- Slide 15
- Math Story Metric Embeddings
- Negative Type Metrics (squared-L2)
- Embedding NEG into L1
- Cuts and L1 Metrics
- Sparsest Cut asymp Optimizing Over L1
- Metric Embeddings amp Sparsity
- Integrality Gap Lower Bound
- Slide 23
- Slide 24
- Recall Integrality Gap Lower Bound
- Slide 26
- Starting Point Hypercube()
- Hypercube hellip
- Cuts in Hypercube Coordinate
- Cuts in Hypercube hellip
- Kahn-Kalai-Linial
- Increasing Size of Balanced Cuts
- eg 4-dim hypercube
- More Formally hellip
- Slide 35
- SDP Solution
- Formally SDP Solution
- Conclusion
-
Metric Embeddings amp Sparsity
bull Optimizing over cuts asymp Optimizing over L1 metrics
bull SDP solution asymp Optimizing over NEG
bull Goemans-LinialARV Conjecture NEG embeds in L1 with O(1) distortion Integrality Gap is O(1)
bull Implies O(1) approx algo for estimating sparsity
Integrality Gap Lower Bound
Sparsest Cut Balanced Separator
log log n integrality gap instance
Khot-Vishnoi rsquo05 Krauthgamer-Rabani lsquo06Devanur-Khot-Saket-Vishnoi lsquo06
bull Disproves the GLARV Conjecture bull Previous best lower bound 116 [Zatloukal rsquo04]
Outline of this Talk Part I
1 Graph PartitioningMotivation amp HistorySDP Approach
2 Embedding Negative Type Metrics into L1 Metric Spaces and EmbeddabilityCut Cone asymp L1
Negative Type Metrics as SDP SolutionsLLRAR Connection Part II
1 Integrality Gap instance Hypercube and Cuts
bull Kahn-Kalai-Linial (isoperimetry of hypercube)The GraphThe SDP Solution
2 Conclusion
Outline of this Talk Part I
1 Graph PartitioningMotivation amp HistorySDP Approach
2 Embedding Negative Type Metrics into L1 Metric Spaces and EmbeddabilityCut Cone asymp L1
Negative Type Metrics as SDP SolutionsLLRAR Connection Part II
1 Integrality Gap instance Hypercube and Cuts
bull Kahn-Kalai-Linial (isoperimetry of hypercube)The GraphThe SDP Solution
2 Conclusion
Recall Integrality Gap Lower Bound
Sparsest Cut Balanced Separator
log log n integrality gap instance
Khot-Vishnoi rsquo05 Krauthgamer-Rabani rsquo06 Devanur-Khot-Saket-Vishnoi lsquo06
bull Disproves the GLARV Conjecture bull Previous best lower bound 116 [Zatloukal rsquo04]
Integrality Gap Lower Bound
Thm Construct graph G(1hellipnE) and unit vector assignment i -gt vi є Rn st
1 G is an ldquoexpanderrdquo every frac14 balanced cut has Ω(|E|(log log n)log n) edges (Kahn-Kalai-Linial)
2 ldquoLowrdquo SDP solution O(|E|log n)
3 Well-Separatedness Σiltj ||vi-vj||2 = n2
4 Triangle inequality d(ij)=||vi-vj||2 is a metric
Integrality gap Ω(log log n)
Starting Point Hypercube()
H=-11k
n = 2k
(111)
(-111)
(11-1)
(1-11)
(1-1-1)
(-1-11)
(-11-1) (-1-1-1)
Hypercube hellip H=-11k Advantages
bull Understand cuts in H tools from Fourier Analysis
bull Vertex is a vector in Rk starting point for SDP solution
But hellip
bull Hypercube has ldquosmallrdquo balanced cuts coordinate cuts have 1k fraction of the edges
Cuts in Hypercube Coordinate of edges = |E(H)|k
Edges across pairs of vertices differing in i-th bit
Cuts in Hypercube hellip
decompose into coordinate cuts
any balanced cut has a coord cut which contributes E(H)k2 edges
Kahn-Kalai-Linial
any balanced cut has a coordinate cut which contributes E(H) (log
k)k2 edges
Increasing Size of Balanced Cuts
consider balanced cuts in which coordinates are indistinguishable
(wrt to their contribution to the cut)
can be achieved by symmetrizing the hypercube
each coordinate contributes equally total E(H) (log k)k edges
eg 4-dim hypercube
(1111)
(-1-1-1-1)
(111-1) (-1111) (1-111) (11-11)
(-1-1-11) (1-1-1-1) (-11-1-1) (-1-11-1)
(11-1-1) (-111-1) (-1-111) (1-1-11)
(1-11-1) (-11-11)
More Formally hellipH=-11k with a rotation group acting on its coordinates
Partitions H into equivalence classes V1hellipVn
Each Vi is a vertex Edges are hypercube edges
G(VE) |E(G)|=|E(H)| k ~ log n
Balanced cuts in G correspond to balanced cuts in H
KKL any balanced cut (in H) has ldquoardquo coordinate cut which contributes E(H) (log k)k2 edges to the cut
Group is transitive ldquoeveryrdquo coordinate cut has the same contribution
Any balanced cut in G has ge E(G) (log k)k = E(G) (log log n)(log n)
Integrality Gap Lower Bound
Thm Construct graph G(1hellipnE) and unit vector assignment i -gt vi є Rn st
1 G is an ldquoexpanderrdquo every frac14 balanced cut has Ω(|E|(log log n)log n) edges (Kahn-Kalai-Linial)
2 ldquoLowrdquo SDP solution O(|E|log n)
3 Well-Separatedness Σiltj ||vi-vj||2 = n2
4 Triangle inequality d(ij)=||vi-vj||2 is a metric
Integrality gap Ω(log log n)
SDP Solution
(1111)
(-1-1-1-1)
(111-1) (-1111) (1-111) (11-11)
(-1-1-11) (1-1-1-1) (-11-1-1) (-1-11-1)
(11-1-1) (-111-1) (-1-111) (1-1-11)
(1-11-1) (-11-11)
Formally SDP SolutionVertex equivalence class x1 x2 hellip xk (rotations)
Vector (1radick) Σj xj
Observations
bull Edge across two nodes differing in one bit
Contribution to sdp ~ 1k sdp le |E(G)|k = |E(G)|(log n)
Triangle Inequality (little bit of work and case analysis)
For most classes x1 hellip xk is ldquonearly orthogonalrdquo
Hidden Gram-Schmidt Orthogonalization Tensoring Well-separatedness
Conclusion
bull (Simple) log log n integrality gap for SCBS
bull Close the gap between log log n and log n
bull [Lee-Naor rsquo06 Cheeger-Kleiner rsquo06] Another counter-example May give (log n)c
Thank you
- Slide 1
- CS Story Sparsity
- Applications
- Estimating Sparsity
- History
- Outline of this Talk
- Slide 7
- Quadratic Program for BS
- SDP for Balanced Separator
- Why is this a Relaxation
- Quadratic Program for BS hellip
- SDP for Balanced Separator hellip
- Geometry of Triangle Ineq
- Integrality Gap Upper Bound
- Slide 15
- Math Story Metric Embeddings
- Negative Type Metrics (squared-L2)
- Embedding NEG into L1
- Cuts and L1 Metrics
- Sparsest Cut asymp Optimizing Over L1
- Metric Embeddings amp Sparsity
- Integrality Gap Lower Bound
- Slide 23
- Slide 24
- Recall Integrality Gap Lower Bound
- Slide 26
- Starting Point Hypercube()
- Hypercube hellip
- Cuts in Hypercube Coordinate
- Cuts in Hypercube hellip
- Kahn-Kalai-Linial
- Increasing Size of Balanced Cuts
- eg 4-dim hypercube
- More Formally hellip
- Slide 35
- SDP Solution
- Formally SDP Solution
- Conclusion
-
Integrality Gap Lower Bound
Sparsest Cut Balanced Separator
log log n integrality gap instance
Khot-Vishnoi rsquo05 Krauthgamer-Rabani lsquo06Devanur-Khot-Saket-Vishnoi lsquo06
bull Disproves the GLARV Conjecture bull Previous best lower bound 116 [Zatloukal rsquo04]
Outline of this Talk Part I
1 Graph PartitioningMotivation amp HistorySDP Approach
2 Embedding Negative Type Metrics into L1 Metric Spaces and EmbeddabilityCut Cone asymp L1
Negative Type Metrics as SDP SolutionsLLRAR Connection Part II
1 Integrality Gap instance Hypercube and Cuts
bull Kahn-Kalai-Linial (isoperimetry of hypercube)The GraphThe SDP Solution
2 Conclusion
Outline of this Talk Part I
1 Graph PartitioningMotivation amp HistorySDP Approach
2 Embedding Negative Type Metrics into L1 Metric Spaces and EmbeddabilityCut Cone asymp L1
Negative Type Metrics as SDP SolutionsLLRAR Connection Part II
1 Integrality Gap instance Hypercube and Cuts
bull Kahn-Kalai-Linial (isoperimetry of hypercube)The GraphThe SDP Solution
2 Conclusion
Recall Integrality Gap Lower Bound
Sparsest Cut Balanced Separator
log log n integrality gap instance
Khot-Vishnoi rsquo05 Krauthgamer-Rabani rsquo06 Devanur-Khot-Saket-Vishnoi lsquo06
bull Disproves the GLARV Conjecture bull Previous best lower bound 116 [Zatloukal rsquo04]
Integrality Gap Lower Bound
Thm Construct graph G(1hellipnE) and unit vector assignment i -gt vi є Rn st
1 G is an ldquoexpanderrdquo every frac14 balanced cut has Ω(|E|(log log n)log n) edges (Kahn-Kalai-Linial)
2 ldquoLowrdquo SDP solution O(|E|log n)
3 Well-Separatedness Σiltj ||vi-vj||2 = n2
4 Triangle inequality d(ij)=||vi-vj||2 is a metric
Integrality gap Ω(log log n)
Starting Point Hypercube()
H=-11k
n = 2k
(111)
(-111)
(11-1)
(1-11)
(1-1-1)
(-1-11)
(-11-1) (-1-1-1)
Hypercube hellip H=-11k Advantages
bull Understand cuts in H tools from Fourier Analysis
bull Vertex is a vector in Rk starting point for SDP solution
But hellip
bull Hypercube has ldquosmallrdquo balanced cuts coordinate cuts have 1k fraction of the edges
Cuts in Hypercube Coordinate of edges = |E(H)|k
Edges across pairs of vertices differing in i-th bit
Cuts in Hypercube hellip
decompose into coordinate cuts
any balanced cut has a coord cut which contributes E(H)k2 edges
Kahn-Kalai-Linial
any balanced cut has a coordinate cut which contributes E(H) (log
k)k2 edges
Increasing Size of Balanced Cuts
consider balanced cuts in which coordinates are indistinguishable
(wrt to their contribution to the cut)
can be achieved by symmetrizing the hypercube
each coordinate contributes equally total E(H) (log k)k edges
eg 4-dim hypercube
(1111)
(-1-1-1-1)
(111-1) (-1111) (1-111) (11-11)
(-1-1-11) (1-1-1-1) (-11-1-1) (-1-11-1)
(11-1-1) (-111-1) (-1-111) (1-1-11)
(1-11-1) (-11-11)
More Formally hellipH=-11k with a rotation group acting on its coordinates
Partitions H into equivalence classes V1hellipVn
Each Vi is a vertex Edges are hypercube edges
G(VE) |E(G)|=|E(H)| k ~ log n
Balanced cuts in G correspond to balanced cuts in H
KKL any balanced cut (in H) has ldquoardquo coordinate cut which contributes E(H) (log k)k2 edges to the cut
Group is transitive ldquoeveryrdquo coordinate cut has the same contribution
Any balanced cut in G has ge E(G) (log k)k = E(G) (log log n)(log n)
Integrality Gap Lower Bound
Thm Construct graph G(1hellipnE) and unit vector assignment i -gt vi є Rn st
1 G is an ldquoexpanderrdquo every frac14 balanced cut has Ω(|E|(log log n)log n) edges (Kahn-Kalai-Linial)
2 ldquoLowrdquo SDP solution O(|E|log n)
3 Well-Separatedness Σiltj ||vi-vj||2 = n2
4 Triangle inequality d(ij)=||vi-vj||2 is a metric
Integrality gap Ω(log log n)
SDP Solution
(1111)
(-1-1-1-1)
(111-1) (-1111) (1-111) (11-11)
(-1-1-11) (1-1-1-1) (-11-1-1) (-1-11-1)
(11-1-1) (-111-1) (-1-111) (1-1-11)
(1-11-1) (-11-11)
Formally SDP SolutionVertex equivalence class x1 x2 hellip xk (rotations)
Vector (1radick) Σj xj
Observations
bull Edge across two nodes differing in one bit
Contribution to sdp ~ 1k sdp le |E(G)|k = |E(G)|(log n)
Triangle Inequality (little bit of work and case analysis)
For most classes x1 hellip xk is ldquonearly orthogonalrdquo
Hidden Gram-Schmidt Orthogonalization Tensoring Well-separatedness
Conclusion
bull (Simple) log log n integrality gap for SCBS
bull Close the gap between log log n and log n
bull [Lee-Naor rsquo06 Cheeger-Kleiner rsquo06] Another counter-example May give (log n)c
Thank you
- Slide 1
- CS Story Sparsity
- Applications
- Estimating Sparsity
- History
- Outline of this Talk
- Slide 7
- Quadratic Program for BS
- SDP for Balanced Separator
- Why is this a Relaxation
- Quadratic Program for BS hellip
- SDP for Balanced Separator hellip
- Geometry of Triangle Ineq
- Integrality Gap Upper Bound
- Slide 15
- Math Story Metric Embeddings
- Negative Type Metrics (squared-L2)
- Embedding NEG into L1
- Cuts and L1 Metrics
- Sparsest Cut asymp Optimizing Over L1
- Metric Embeddings amp Sparsity
- Integrality Gap Lower Bound
- Slide 23
- Slide 24
- Recall Integrality Gap Lower Bound
- Slide 26
- Starting Point Hypercube()
- Hypercube hellip
- Cuts in Hypercube Coordinate
- Cuts in Hypercube hellip
- Kahn-Kalai-Linial
- Increasing Size of Balanced Cuts
- eg 4-dim hypercube
- More Formally hellip
- Slide 35
- SDP Solution
- Formally SDP Solution
- Conclusion
-
Outline of this Talk Part I
1 Graph PartitioningMotivation amp HistorySDP Approach
2 Embedding Negative Type Metrics into L1 Metric Spaces and EmbeddabilityCut Cone asymp L1
Negative Type Metrics as SDP SolutionsLLRAR Connection Part II
1 Integrality Gap instance Hypercube and Cuts
bull Kahn-Kalai-Linial (isoperimetry of hypercube)The GraphThe SDP Solution
2 Conclusion
Outline of this Talk Part I
1 Graph PartitioningMotivation amp HistorySDP Approach
2 Embedding Negative Type Metrics into L1 Metric Spaces and EmbeddabilityCut Cone asymp L1
Negative Type Metrics as SDP SolutionsLLRAR Connection Part II
1 Integrality Gap instance Hypercube and Cuts
bull Kahn-Kalai-Linial (isoperimetry of hypercube)The GraphThe SDP Solution
2 Conclusion
Recall Integrality Gap Lower Bound
Sparsest Cut Balanced Separator
log log n integrality gap instance
Khot-Vishnoi rsquo05 Krauthgamer-Rabani rsquo06 Devanur-Khot-Saket-Vishnoi lsquo06
bull Disproves the GLARV Conjecture bull Previous best lower bound 116 [Zatloukal rsquo04]
Integrality Gap Lower Bound
Thm Construct graph G(1hellipnE) and unit vector assignment i -gt vi є Rn st
1 G is an ldquoexpanderrdquo every frac14 balanced cut has Ω(|E|(log log n)log n) edges (Kahn-Kalai-Linial)
2 ldquoLowrdquo SDP solution O(|E|log n)
3 Well-Separatedness Σiltj ||vi-vj||2 = n2
4 Triangle inequality d(ij)=||vi-vj||2 is a metric
Integrality gap Ω(log log n)
Starting Point Hypercube()
H=-11k
n = 2k
(111)
(-111)
(11-1)
(1-11)
(1-1-1)
(-1-11)
(-11-1) (-1-1-1)
Hypercube hellip H=-11k Advantages
bull Understand cuts in H tools from Fourier Analysis
bull Vertex is a vector in Rk starting point for SDP solution
But hellip
bull Hypercube has ldquosmallrdquo balanced cuts coordinate cuts have 1k fraction of the edges
Cuts in Hypercube Coordinate of edges = |E(H)|k
Edges across pairs of vertices differing in i-th bit
Cuts in Hypercube hellip
decompose into coordinate cuts
any balanced cut has a coord cut which contributes E(H)k2 edges
Kahn-Kalai-Linial
any balanced cut has a coordinate cut which contributes E(H) (log
k)k2 edges
Increasing Size of Balanced Cuts
consider balanced cuts in which coordinates are indistinguishable
(wrt to their contribution to the cut)
can be achieved by symmetrizing the hypercube
each coordinate contributes equally total E(H) (log k)k edges
eg 4-dim hypercube
(1111)
(-1-1-1-1)
(111-1) (-1111) (1-111) (11-11)
(-1-1-11) (1-1-1-1) (-11-1-1) (-1-11-1)
(11-1-1) (-111-1) (-1-111) (1-1-11)
(1-11-1) (-11-11)
More Formally hellipH=-11k with a rotation group acting on its coordinates
Partitions H into equivalence classes V1hellipVn
Each Vi is a vertex Edges are hypercube edges
G(VE) |E(G)|=|E(H)| k ~ log n
Balanced cuts in G correspond to balanced cuts in H
KKL any balanced cut (in H) has ldquoardquo coordinate cut which contributes E(H) (log k)k2 edges to the cut
Group is transitive ldquoeveryrdquo coordinate cut has the same contribution
Any balanced cut in G has ge E(G) (log k)k = E(G) (log log n)(log n)
Integrality Gap Lower Bound
Thm Construct graph G(1hellipnE) and unit vector assignment i -gt vi є Rn st
1 G is an ldquoexpanderrdquo every frac14 balanced cut has Ω(|E|(log log n)log n) edges (Kahn-Kalai-Linial)
2 ldquoLowrdquo SDP solution O(|E|log n)
3 Well-Separatedness Σiltj ||vi-vj||2 = n2
4 Triangle inequality d(ij)=||vi-vj||2 is a metric
Integrality gap Ω(log log n)
SDP Solution
(1111)
(-1-1-1-1)
(111-1) (-1111) (1-111) (11-11)
(-1-1-11) (1-1-1-1) (-11-1-1) (-1-11-1)
(11-1-1) (-111-1) (-1-111) (1-1-11)
(1-11-1) (-11-11)
Formally SDP SolutionVertex equivalence class x1 x2 hellip xk (rotations)
Vector (1radick) Σj xj
Observations
bull Edge across two nodes differing in one bit
Contribution to sdp ~ 1k sdp le |E(G)|k = |E(G)|(log n)
Triangle Inequality (little bit of work and case analysis)
For most classes x1 hellip xk is ldquonearly orthogonalrdquo
Hidden Gram-Schmidt Orthogonalization Tensoring Well-separatedness
Conclusion
bull (Simple) log log n integrality gap for SCBS
bull Close the gap between log log n and log n
bull [Lee-Naor rsquo06 Cheeger-Kleiner rsquo06] Another counter-example May give (log n)c
Thank you
- Slide 1
- CS Story Sparsity
- Applications
- Estimating Sparsity
- History
- Outline of this Talk
- Slide 7
- Quadratic Program for BS
- SDP for Balanced Separator
- Why is this a Relaxation
- Quadratic Program for BS hellip
- SDP for Balanced Separator hellip
- Geometry of Triangle Ineq
- Integrality Gap Upper Bound
- Slide 15
- Math Story Metric Embeddings
- Negative Type Metrics (squared-L2)
- Embedding NEG into L1
- Cuts and L1 Metrics
- Sparsest Cut asymp Optimizing Over L1
- Metric Embeddings amp Sparsity
- Integrality Gap Lower Bound
- Slide 23
- Slide 24
- Recall Integrality Gap Lower Bound
- Slide 26
- Starting Point Hypercube()
- Hypercube hellip
- Cuts in Hypercube Coordinate
- Cuts in Hypercube hellip
- Kahn-Kalai-Linial
- Increasing Size of Balanced Cuts
- eg 4-dim hypercube
- More Formally hellip
- Slide 35
- SDP Solution
- Formally SDP Solution
- Conclusion
-
Outline of this Talk Part I
1 Graph PartitioningMotivation amp HistorySDP Approach
2 Embedding Negative Type Metrics into L1 Metric Spaces and EmbeddabilityCut Cone asymp L1
Negative Type Metrics as SDP SolutionsLLRAR Connection Part II
1 Integrality Gap instance Hypercube and Cuts
bull Kahn-Kalai-Linial (isoperimetry of hypercube)The GraphThe SDP Solution
2 Conclusion
Recall Integrality Gap Lower Bound
Sparsest Cut Balanced Separator
log log n integrality gap instance
Khot-Vishnoi rsquo05 Krauthgamer-Rabani rsquo06 Devanur-Khot-Saket-Vishnoi lsquo06
bull Disproves the GLARV Conjecture bull Previous best lower bound 116 [Zatloukal rsquo04]
Integrality Gap Lower Bound
Thm Construct graph G(1hellipnE) and unit vector assignment i -gt vi є Rn st
1 G is an ldquoexpanderrdquo every frac14 balanced cut has Ω(|E|(log log n)log n) edges (Kahn-Kalai-Linial)
2 ldquoLowrdquo SDP solution O(|E|log n)
3 Well-Separatedness Σiltj ||vi-vj||2 = n2
4 Triangle inequality d(ij)=||vi-vj||2 is a metric
Integrality gap Ω(log log n)
Starting Point Hypercube()
H=-11k
n = 2k
(111)
(-111)
(11-1)
(1-11)
(1-1-1)
(-1-11)
(-11-1) (-1-1-1)
Hypercube hellip H=-11k Advantages
bull Understand cuts in H tools from Fourier Analysis
bull Vertex is a vector in Rk starting point for SDP solution
But hellip
bull Hypercube has ldquosmallrdquo balanced cuts coordinate cuts have 1k fraction of the edges
Cuts in Hypercube Coordinate of edges = |E(H)|k
Edges across pairs of vertices differing in i-th bit
Cuts in Hypercube hellip
decompose into coordinate cuts
any balanced cut has a coord cut which contributes E(H)k2 edges
Kahn-Kalai-Linial
any balanced cut has a coordinate cut which contributes E(H) (log
k)k2 edges
Increasing Size of Balanced Cuts
consider balanced cuts in which coordinates are indistinguishable
(wrt to their contribution to the cut)
can be achieved by symmetrizing the hypercube
each coordinate contributes equally total E(H) (log k)k edges
eg 4-dim hypercube
(1111)
(-1-1-1-1)
(111-1) (-1111) (1-111) (11-11)
(-1-1-11) (1-1-1-1) (-11-1-1) (-1-11-1)
(11-1-1) (-111-1) (-1-111) (1-1-11)
(1-11-1) (-11-11)
More Formally hellipH=-11k with a rotation group acting on its coordinates
Partitions H into equivalence classes V1hellipVn
Each Vi is a vertex Edges are hypercube edges
G(VE) |E(G)|=|E(H)| k ~ log n
Balanced cuts in G correspond to balanced cuts in H
KKL any balanced cut (in H) has ldquoardquo coordinate cut which contributes E(H) (log k)k2 edges to the cut
Group is transitive ldquoeveryrdquo coordinate cut has the same contribution
Any balanced cut in G has ge E(G) (log k)k = E(G) (log log n)(log n)
Integrality Gap Lower Bound
Thm Construct graph G(1hellipnE) and unit vector assignment i -gt vi є Rn st
1 G is an ldquoexpanderrdquo every frac14 balanced cut has Ω(|E|(log log n)log n) edges (Kahn-Kalai-Linial)
2 ldquoLowrdquo SDP solution O(|E|log n)
3 Well-Separatedness Σiltj ||vi-vj||2 = n2
4 Triangle inequality d(ij)=||vi-vj||2 is a metric
Integrality gap Ω(log log n)
SDP Solution
(1111)
(-1-1-1-1)
(111-1) (-1111) (1-111) (11-11)
(-1-1-11) (1-1-1-1) (-11-1-1) (-1-11-1)
(11-1-1) (-111-1) (-1-111) (1-1-11)
(1-11-1) (-11-11)
Formally SDP SolutionVertex equivalence class x1 x2 hellip xk (rotations)
Vector (1radick) Σj xj
Observations
bull Edge across two nodes differing in one bit
Contribution to sdp ~ 1k sdp le |E(G)|k = |E(G)|(log n)
Triangle Inequality (little bit of work and case analysis)
For most classes x1 hellip xk is ldquonearly orthogonalrdquo
Hidden Gram-Schmidt Orthogonalization Tensoring Well-separatedness
Conclusion
bull (Simple) log log n integrality gap for SCBS
bull Close the gap between log log n and log n
bull [Lee-Naor rsquo06 Cheeger-Kleiner rsquo06] Another counter-example May give (log n)c
Thank you
- Slide 1
- CS Story Sparsity
- Applications
- Estimating Sparsity
- History
- Outline of this Talk
- Slide 7
- Quadratic Program for BS
- SDP for Balanced Separator
- Why is this a Relaxation
- Quadratic Program for BS hellip
- SDP for Balanced Separator hellip
- Geometry of Triangle Ineq
- Integrality Gap Upper Bound
- Slide 15
- Math Story Metric Embeddings
- Negative Type Metrics (squared-L2)
- Embedding NEG into L1
- Cuts and L1 Metrics
- Sparsest Cut asymp Optimizing Over L1
- Metric Embeddings amp Sparsity
- Integrality Gap Lower Bound
- Slide 23
- Slide 24
- Recall Integrality Gap Lower Bound
- Slide 26
- Starting Point Hypercube()
- Hypercube hellip
- Cuts in Hypercube Coordinate
- Cuts in Hypercube hellip
- Kahn-Kalai-Linial
- Increasing Size of Balanced Cuts
- eg 4-dim hypercube
- More Formally hellip
- Slide 35
- SDP Solution
- Formally SDP Solution
- Conclusion
-
Recall Integrality Gap Lower Bound
Sparsest Cut Balanced Separator
log log n integrality gap instance
Khot-Vishnoi rsquo05 Krauthgamer-Rabani rsquo06 Devanur-Khot-Saket-Vishnoi lsquo06
bull Disproves the GLARV Conjecture bull Previous best lower bound 116 [Zatloukal rsquo04]
Integrality Gap Lower Bound
Thm Construct graph G(1hellipnE) and unit vector assignment i -gt vi є Rn st
1 G is an ldquoexpanderrdquo every frac14 balanced cut has Ω(|E|(log log n)log n) edges (Kahn-Kalai-Linial)
2 ldquoLowrdquo SDP solution O(|E|log n)
3 Well-Separatedness Σiltj ||vi-vj||2 = n2
4 Triangle inequality d(ij)=||vi-vj||2 is a metric
Integrality gap Ω(log log n)
Starting Point Hypercube()
H=-11k
n = 2k
(111)
(-111)
(11-1)
(1-11)
(1-1-1)
(-1-11)
(-11-1) (-1-1-1)
Hypercube hellip H=-11k Advantages
bull Understand cuts in H tools from Fourier Analysis
bull Vertex is a vector in Rk starting point for SDP solution
But hellip
bull Hypercube has ldquosmallrdquo balanced cuts coordinate cuts have 1k fraction of the edges
Cuts in Hypercube Coordinate of edges = |E(H)|k
Edges across pairs of vertices differing in i-th bit
Cuts in Hypercube hellip
decompose into coordinate cuts
any balanced cut has a coord cut which contributes E(H)k2 edges
Kahn-Kalai-Linial
any balanced cut has a coordinate cut which contributes E(H) (log
k)k2 edges
Increasing Size of Balanced Cuts
consider balanced cuts in which coordinates are indistinguishable
(wrt to their contribution to the cut)
can be achieved by symmetrizing the hypercube
each coordinate contributes equally total E(H) (log k)k edges
eg 4-dim hypercube
(1111)
(-1-1-1-1)
(111-1) (-1111) (1-111) (11-11)
(-1-1-11) (1-1-1-1) (-11-1-1) (-1-11-1)
(11-1-1) (-111-1) (-1-111) (1-1-11)
(1-11-1) (-11-11)
More Formally hellipH=-11k with a rotation group acting on its coordinates
Partitions H into equivalence classes V1hellipVn
Each Vi is a vertex Edges are hypercube edges
G(VE) |E(G)|=|E(H)| k ~ log n
Balanced cuts in G correspond to balanced cuts in H
KKL any balanced cut (in H) has ldquoardquo coordinate cut which contributes E(H) (log k)k2 edges to the cut
Group is transitive ldquoeveryrdquo coordinate cut has the same contribution
Any balanced cut in G has ge E(G) (log k)k = E(G) (log log n)(log n)
Integrality Gap Lower Bound
Thm Construct graph G(1hellipnE) and unit vector assignment i -gt vi є Rn st
1 G is an ldquoexpanderrdquo every frac14 balanced cut has Ω(|E|(log log n)log n) edges (Kahn-Kalai-Linial)
2 ldquoLowrdquo SDP solution O(|E|log n)
3 Well-Separatedness Σiltj ||vi-vj||2 = n2
4 Triangle inequality d(ij)=||vi-vj||2 is a metric
Integrality gap Ω(log log n)
SDP Solution
(1111)
(-1-1-1-1)
(111-1) (-1111) (1-111) (11-11)
(-1-1-11) (1-1-1-1) (-11-1-1) (-1-11-1)
(11-1-1) (-111-1) (-1-111) (1-1-11)
(1-11-1) (-11-11)
Formally SDP SolutionVertex equivalence class x1 x2 hellip xk (rotations)
Vector (1radick) Σj xj
Observations
bull Edge across two nodes differing in one bit
Contribution to sdp ~ 1k sdp le |E(G)|k = |E(G)|(log n)
Triangle Inequality (little bit of work and case analysis)
For most classes x1 hellip xk is ldquonearly orthogonalrdquo
Hidden Gram-Schmidt Orthogonalization Tensoring Well-separatedness
Conclusion
bull (Simple) log log n integrality gap for SCBS
bull Close the gap between log log n and log n
bull [Lee-Naor rsquo06 Cheeger-Kleiner rsquo06] Another counter-example May give (log n)c
Thank you
- Slide 1
- CS Story Sparsity
- Applications
- Estimating Sparsity
- History
- Outline of this Talk
- Slide 7
- Quadratic Program for BS
- SDP for Balanced Separator
- Why is this a Relaxation
- Quadratic Program for BS hellip
- SDP for Balanced Separator hellip
- Geometry of Triangle Ineq
- Integrality Gap Upper Bound
- Slide 15
- Math Story Metric Embeddings
- Negative Type Metrics (squared-L2)
- Embedding NEG into L1
- Cuts and L1 Metrics
- Sparsest Cut asymp Optimizing Over L1
- Metric Embeddings amp Sparsity
- Integrality Gap Lower Bound
- Slide 23
- Slide 24
- Recall Integrality Gap Lower Bound
- Slide 26
- Starting Point Hypercube()
- Hypercube hellip
- Cuts in Hypercube Coordinate
- Cuts in Hypercube hellip
- Kahn-Kalai-Linial
- Increasing Size of Balanced Cuts
- eg 4-dim hypercube
- More Formally hellip
- Slide 35
- SDP Solution
- Formally SDP Solution
- Conclusion
-
Integrality Gap Lower Bound
Thm Construct graph G(1hellipnE) and unit vector assignment i -gt vi є Rn st
1 G is an ldquoexpanderrdquo every frac14 balanced cut has Ω(|E|(log log n)log n) edges (Kahn-Kalai-Linial)
2 ldquoLowrdquo SDP solution O(|E|log n)
3 Well-Separatedness Σiltj ||vi-vj||2 = n2
4 Triangle inequality d(ij)=||vi-vj||2 is a metric
Integrality gap Ω(log log n)
Starting Point Hypercube()
H=-11k
n = 2k
(111)
(-111)
(11-1)
(1-11)
(1-1-1)
(-1-11)
(-11-1) (-1-1-1)
Hypercube hellip H=-11k Advantages
bull Understand cuts in H tools from Fourier Analysis
bull Vertex is a vector in Rk starting point for SDP solution
But hellip
bull Hypercube has ldquosmallrdquo balanced cuts coordinate cuts have 1k fraction of the edges
Cuts in Hypercube Coordinate of edges = |E(H)|k
Edges across pairs of vertices differing in i-th bit
Cuts in Hypercube hellip
decompose into coordinate cuts
any balanced cut has a coord cut which contributes E(H)k2 edges
Kahn-Kalai-Linial
any balanced cut has a coordinate cut which contributes E(H) (log
k)k2 edges
Increasing Size of Balanced Cuts
consider balanced cuts in which coordinates are indistinguishable
(wrt to their contribution to the cut)
can be achieved by symmetrizing the hypercube
each coordinate contributes equally total E(H) (log k)k edges
eg 4-dim hypercube
(1111)
(-1-1-1-1)
(111-1) (-1111) (1-111) (11-11)
(-1-1-11) (1-1-1-1) (-11-1-1) (-1-11-1)
(11-1-1) (-111-1) (-1-111) (1-1-11)
(1-11-1) (-11-11)
More Formally hellipH=-11k with a rotation group acting on its coordinates
Partitions H into equivalence classes V1hellipVn
Each Vi is a vertex Edges are hypercube edges
G(VE) |E(G)|=|E(H)| k ~ log n
Balanced cuts in G correspond to balanced cuts in H
KKL any balanced cut (in H) has ldquoardquo coordinate cut which contributes E(H) (log k)k2 edges to the cut
Group is transitive ldquoeveryrdquo coordinate cut has the same contribution
Any balanced cut in G has ge E(G) (log k)k = E(G) (log log n)(log n)
Integrality Gap Lower Bound
Thm Construct graph G(1hellipnE) and unit vector assignment i -gt vi є Rn st
1 G is an ldquoexpanderrdquo every frac14 balanced cut has Ω(|E|(log log n)log n) edges (Kahn-Kalai-Linial)
2 ldquoLowrdquo SDP solution O(|E|log n)
3 Well-Separatedness Σiltj ||vi-vj||2 = n2
4 Triangle inequality d(ij)=||vi-vj||2 is a metric
Integrality gap Ω(log log n)
SDP Solution
(1111)
(-1-1-1-1)
(111-1) (-1111) (1-111) (11-11)
(-1-1-11) (1-1-1-1) (-11-1-1) (-1-11-1)
(11-1-1) (-111-1) (-1-111) (1-1-11)
(1-11-1) (-11-11)
Formally SDP SolutionVertex equivalence class x1 x2 hellip xk (rotations)
Vector (1radick) Σj xj
Observations
bull Edge across two nodes differing in one bit
Contribution to sdp ~ 1k sdp le |E(G)|k = |E(G)|(log n)
Triangle Inequality (little bit of work and case analysis)
For most classes x1 hellip xk is ldquonearly orthogonalrdquo
Hidden Gram-Schmidt Orthogonalization Tensoring Well-separatedness
Conclusion
bull (Simple) log log n integrality gap for SCBS
bull Close the gap between log log n and log n
bull [Lee-Naor rsquo06 Cheeger-Kleiner rsquo06] Another counter-example May give (log n)c
Thank you
- Slide 1
- CS Story Sparsity
- Applications
- Estimating Sparsity
- History
- Outline of this Talk
- Slide 7
- Quadratic Program for BS
- SDP for Balanced Separator
- Why is this a Relaxation
- Quadratic Program for BS hellip
- SDP for Balanced Separator hellip
- Geometry of Triangle Ineq
- Integrality Gap Upper Bound
- Slide 15
- Math Story Metric Embeddings
- Negative Type Metrics (squared-L2)
- Embedding NEG into L1
- Cuts and L1 Metrics
- Sparsest Cut asymp Optimizing Over L1
- Metric Embeddings amp Sparsity
- Integrality Gap Lower Bound
- Slide 23
- Slide 24
- Recall Integrality Gap Lower Bound
- Slide 26
- Starting Point Hypercube()
- Hypercube hellip
- Cuts in Hypercube Coordinate
- Cuts in Hypercube hellip
- Kahn-Kalai-Linial
- Increasing Size of Balanced Cuts
- eg 4-dim hypercube
- More Formally hellip
- Slide 35
- SDP Solution
- Formally SDP Solution
- Conclusion
-
Starting Point Hypercube()
H=-11k
n = 2k
(111)
(-111)
(11-1)
(1-11)
(1-1-1)
(-1-11)
(-11-1) (-1-1-1)
Hypercube hellip H=-11k Advantages
bull Understand cuts in H tools from Fourier Analysis
bull Vertex is a vector in Rk starting point for SDP solution
But hellip
bull Hypercube has ldquosmallrdquo balanced cuts coordinate cuts have 1k fraction of the edges
Cuts in Hypercube Coordinate of edges = |E(H)|k
Edges across pairs of vertices differing in i-th bit
Cuts in Hypercube hellip
decompose into coordinate cuts
any balanced cut has a coord cut which contributes E(H)k2 edges
Kahn-Kalai-Linial
any balanced cut has a coordinate cut which contributes E(H) (log
k)k2 edges
Increasing Size of Balanced Cuts
consider balanced cuts in which coordinates are indistinguishable
(wrt to their contribution to the cut)
can be achieved by symmetrizing the hypercube
each coordinate contributes equally total E(H) (log k)k edges
eg 4-dim hypercube
(1111)
(-1-1-1-1)
(111-1) (-1111) (1-111) (11-11)
(-1-1-11) (1-1-1-1) (-11-1-1) (-1-11-1)
(11-1-1) (-111-1) (-1-111) (1-1-11)
(1-11-1) (-11-11)
More Formally hellipH=-11k with a rotation group acting on its coordinates
Partitions H into equivalence classes V1hellipVn
Each Vi is a vertex Edges are hypercube edges
G(VE) |E(G)|=|E(H)| k ~ log n
Balanced cuts in G correspond to balanced cuts in H
KKL any balanced cut (in H) has ldquoardquo coordinate cut which contributes E(H) (log k)k2 edges to the cut
Group is transitive ldquoeveryrdquo coordinate cut has the same contribution
Any balanced cut in G has ge E(G) (log k)k = E(G) (log log n)(log n)
Integrality Gap Lower Bound
Thm Construct graph G(1hellipnE) and unit vector assignment i -gt vi є Rn st
1 G is an ldquoexpanderrdquo every frac14 balanced cut has Ω(|E|(log log n)log n) edges (Kahn-Kalai-Linial)
2 ldquoLowrdquo SDP solution O(|E|log n)
3 Well-Separatedness Σiltj ||vi-vj||2 = n2
4 Triangle inequality d(ij)=||vi-vj||2 is a metric
Integrality gap Ω(log log n)
SDP Solution
(1111)
(-1-1-1-1)
(111-1) (-1111) (1-111) (11-11)
(-1-1-11) (1-1-1-1) (-11-1-1) (-1-11-1)
(11-1-1) (-111-1) (-1-111) (1-1-11)
(1-11-1) (-11-11)
Formally SDP SolutionVertex equivalence class x1 x2 hellip xk (rotations)
Vector (1radick) Σj xj
Observations
bull Edge across two nodes differing in one bit
Contribution to sdp ~ 1k sdp le |E(G)|k = |E(G)|(log n)
Triangle Inequality (little bit of work and case analysis)
For most classes x1 hellip xk is ldquonearly orthogonalrdquo
Hidden Gram-Schmidt Orthogonalization Tensoring Well-separatedness
Conclusion
bull (Simple) log log n integrality gap for SCBS
bull Close the gap between log log n and log n
bull [Lee-Naor rsquo06 Cheeger-Kleiner rsquo06] Another counter-example May give (log n)c
Thank you
- Slide 1
- CS Story Sparsity
- Applications
- Estimating Sparsity
- History
- Outline of this Talk
- Slide 7
- Quadratic Program for BS
- SDP for Balanced Separator
- Why is this a Relaxation
- Quadratic Program for BS hellip
- SDP for Balanced Separator hellip
- Geometry of Triangle Ineq
- Integrality Gap Upper Bound
- Slide 15
- Math Story Metric Embeddings
- Negative Type Metrics (squared-L2)
- Embedding NEG into L1
- Cuts and L1 Metrics
- Sparsest Cut asymp Optimizing Over L1
- Metric Embeddings amp Sparsity
- Integrality Gap Lower Bound
- Slide 23
- Slide 24
- Recall Integrality Gap Lower Bound
- Slide 26
- Starting Point Hypercube()
- Hypercube hellip
- Cuts in Hypercube Coordinate
- Cuts in Hypercube hellip
- Kahn-Kalai-Linial
- Increasing Size of Balanced Cuts
- eg 4-dim hypercube
- More Formally hellip
- Slide 35
- SDP Solution
- Formally SDP Solution
- Conclusion
-
Hypercube hellip H=-11k Advantages
bull Understand cuts in H tools from Fourier Analysis
bull Vertex is a vector in Rk starting point for SDP solution
But hellip
bull Hypercube has ldquosmallrdquo balanced cuts coordinate cuts have 1k fraction of the edges
Cuts in Hypercube Coordinate of edges = |E(H)|k
Edges across pairs of vertices differing in i-th bit
Cuts in Hypercube hellip
decompose into coordinate cuts
any balanced cut has a coord cut which contributes E(H)k2 edges
Kahn-Kalai-Linial
any balanced cut has a coordinate cut which contributes E(H) (log
k)k2 edges
Increasing Size of Balanced Cuts
consider balanced cuts in which coordinates are indistinguishable
(wrt to their contribution to the cut)
can be achieved by symmetrizing the hypercube
each coordinate contributes equally total E(H) (log k)k edges
eg 4-dim hypercube
(1111)
(-1-1-1-1)
(111-1) (-1111) (1-111) (11-11)
(-1-1-11) (1-1-1-1) (-11-1-1) (-1-11-1)
(11-1-1) (-111-1) (-1-111) (1-1-11)
(1-11-1) (-11-11)
More Formally hellipH=-11k with a rotation group acting on its coordinates
Partitions H into equivalence classes V1hellipVn
Each Vi is a vertex Edges are hypercube edges
G(VE) |E(G)|=|E(H)| k ~ log n
Balanced cuts in G correspond to balanced cuts in H
KKL any balanced cut (in H) has ldquoardquo coordinate cut which contributes E(H) (log k)k2 edges to the cut
Group is transitive ldquoeveryrdquo coordinate cut has the same contribution
Any balanced cut in G has ge E(G) (log k)k = E(G) (log log n)(log n)
Integrality Gap Lower Bound
Thm Construct graph G(1hellipnE) and unit vector assignment i -gt vi є Rn st
1 G is an ldquoexpanderrdquo every frac14 balanced cut has Ω(|E|(log log n)log n) edges (Kahn-Kalai-Linial)
2 ldquoLowrdquo SDP solution O(|E|log n)
3 Well-Separatedness Σiltj ||vi-vj||2 = n2
4 Triangle inequality d(ij)=||vi-vj||2 is a metric
Integrality gap Ω(log log n)
SDP Solution
(1111)
(-1-1-1-1)
(111-1) (-1111) (1-111) (11-11)
(-1-1-11) (1-1-1-1) (-11-1-1) (-1-11-1)
(11-1-1) (-111-1) (-1-111) (1-1-11)
(1-11-1) (-11-11)
Formally SDP SolutionVertex equivalence class x1 x2 hellip xk (rotations)
Vector (1radick) Σj xj
Observations
bull Edge across two nodes differing in one bit
Contribution to sdp ~ 1k sdp le |E(G)|k = |E(G)|(log n)
Triangle Inequality (little bit of work and case analysis)
For most classes x1 hellip xk is ldquonearly orthogonalrdquo
Hidden Gram-Schmidt Orthogonalization Tensoring Well-separatedness
Conclusion
bull (Simple) log log n integrality gap for SCBS
bull Close the gap between log log n and log n
bull [Lee-Naor rsquo06 Cheeger-Kleiner rsquo06] Another counter-example May give (log n)c
Thank you
- Slide 1
- CS Story Sparsity
- Applications
- Estimating Sparsity
- History
- Outline of this Talk
- Slide 7
- Quadratic Program for BS
- SDP for Balanced Separator
- Why is this a Relaxation
- Quadratic Program for BS hellip
- SDP for Balanced Separator hellip
- Geometry of Triangle Ineq
- Integrality Gap Upper Bound
- Slide 15
- Math Story Metric Embeddings
- Negative Type Metrics (squared-L2)
- Embedding NEG into L1
- Cuts and L1 Metrics
- Sparsest Cut asymp Optimizing Over L1
- Metric Embeddings amp Sparsity
- Integrality Gap Lower Bound
- Slide 23
- Slide 24
- Recall Integrality Gap Lower Bound
- Slide 26
- Starting Point Hypercube()
- Hypercube hellip
- Cuts in Hypercube Coordinate
- Cuts in Hypercube hellip
- Kahn-Kalai-Linial
- Increasing Size of Balanced Cuts
- eg 4-dim hypercube
- More Formally hellip
- Slide 35
- SDP Solution
- Formally SDP Solution
- Conclusion
-
Cuts in Hypercube Coordinate of edges = |E(H)|k
Edges across pairs of vertices differing in i-th bit
Cuts in Hypercube hellip
decompose into coordinate cuts
any balanced cut has a coord cut which contributes E(H)k2 edges
Kahn-Kalai-Linial
any balanced cut has a coordinate cut which contributes E(H) (log
k)k2 edges
Increasing Size of Balanced Cuts
consider balanced cuts in which coordinates are indistinguishable
(wrt to their contribution to the cut)
can be achieved by symmetrizing the hypercube
each coordinate contributes equally total E(H) (log k)k edges
eg 4-dim hypercube
(1111)
(-1-1-1-1)
(111-1) (-1111) (1-111) (11-11)
(-1-1-11) (1-1-1-1) (-11-1-1) (-1-11-1)
(11-1-1) (-111-1) (-1-111) (1-1-11)
(1-11-1) (-11-11)
More Formally hellipH=-11k with a rotation group acting on its coordinates
Partitions H into equivalence classes V1hellipVn
Each Vi is a vertex Edges are hypercube edges
G(VE) |E(G)|=|E(H)| k ~ log n
Balanced cuts in G correspond to balanced cuts in H
KKL any balanced cut (in H) has ldquoardquo coordinate cut which contributes E(H) (log k)k2 edges to the cut
Group is transitive ldquoeveryrdquo coordinate cut has the same contribution
Any balanced cut in G has ge E(G) (log k)k = E(G) (log log n)(log n)
Integrality Gap Lower Bound
Thm Construct graph G(1hellipnE) and unit vector assignment i -gt vi є Rn st
1 G is an ldquoexpanderrdquo every frac14 balanced cut has Ω(|E|(log log n)log n) edges (Kahn-Kalai-Linial)
2 ldquoLowrdquo SDP solution O(|E|log n)
3 Well-Separatedness Σiltj ||vi-vj||2 = n2
4 Triangle inequality d(ij)=||vi-vj||2 is a metric
Integrality gap Ω(log log n)
SDP Solution
(1111)
(-1-1-1-1)
(111-1) (-1111) (1-111) (11-11)
(-1-1-11) (1-1-1-1) (-11-1-1) (-1-11-1)
(11-1-1) (-111-1) (-1-111) (1-1-11)
(1-11-1) (-11-11)
Formally SDP SolutionVertex equivalence class x1 x2 hellip xk (rotations)
Vector (1radick) Σj xj
Observations
bull Edge across two nodes differing in one bit
Contribution to sdp ~ 1k sdp le |E(G)|k = |E(G)|(log n)
Triangle Inequality (little bit of work and case analysis)
For most classes x1 hellip xk is ldquonearly orthogonalrdquo
Hidden Gram-Schmidt Orthogonalization Tensoring Well-separatedness
Conclusion
bull (Simple) log log n integrality gap for SCBS
bull Close the gap between log log n and log n
bull [Lee-Naor rsquo06 Cheeger-Kleiner rsquo06] Another counter-example May give (log n)c
Thank you
- Slide 1
- CS Story Sparsity
- Applications
- Estimating Sparsity
- History
- Outline of this Talk
- Slide 7
- Quadratic Program for BS
- SDP for Balanced Separator
- Why is this a Relaxation
- Quadratic Program for BS hellip
- SDP for Balanced Separator hellip
- Geometry of Triangle Ineq
- Integrality Gap Upper Bound
- Slide 15
- Math Story Metric Embeddings
- Negative Type Metrics (squared-L2)
- Embedding NEG into L1
- Cuts and L1 Metrics
- Sparsest Cut asymp Optimizing Over L1
- Metric Embeddings amp Sparsity
- Integrality Gap Lower Bound
- Slide 23
- Slide 24
- Recall Integrality Gap Lower Bound
- Slide 26
- Starting Point Hypercube()
- Hypercube hellip
- Cuts in Hypercube Coordinate
- Cuts in Hypercube hellip
- Kahn-Kalai-Linial
- Increasing Size of Balanced Cuts
- eg 4-dim hypercube
- More Formally hellip
- Slide 35
- SDP Solution
- Formally SDP Solution
- Conclusion
-
Cuts in Hypercube hellip
decompose into coordinate cuts
any balanced cut has a coord cut which contributes E(H)k2 edges
Kahn-Kalai-Linial
any balanced cut has a coordinate cut which contributes E(H) (log
k)k2 edges
Increasing Size of Balanced Cuts
consider balanced cuts in which coordinates are indistinguishable
(wrt to their contribution to the cut)
can be achieved by symmetrizing the hypercube
each coordinate contributes equally total E(H) (log k)k edges
eg 4-dim hypercube
(1111)
(-1-1-1-1)
(111-1) (-1111) (1-111) (11-11)
(-1-1-11) (1-1-1-1) (-11-1-1) (-1-11-1)
(11-1-1) (-111-1) (-1-111) (1-1-11)
(1-11-1) (-11-11)
More Formally hellipH=-11k with a rotation group acting on its coordinates
Partitions H into equivalence classes V1hellipVn
Each Vi is a vertex Edges are hypercube edges
G(VE) |E(G)|=|E(H)| k ~ log n
Balanced cuts in G correspond to balanced cuts in H
KKL any balanced cut (in H) has ldquoardquo coordinate cut which contributes E(H) (log k)k2 edges to the cut
Group is transitive ldquoeveryrdquo coordinate cut has the same contribution
Any balanced cut in G has ge E(G) (log k)k = E(G) (log log n)(log n)
Integrality Gap Lower Bound
Thm Construct graph G(1hellipnE) and unit vector assignment i -gt vi є Rn st
1 G is an ldquoexpanderrdquo every frac14 balanced cut has Ω(|E|(log log n)log n) edges (Kahn-Kalai-Linial)
2 ldquoLowrdquo SDP solution O(|E|log n)
3 Well-Separatedness Σiltj ||vi-vj||2 = n2
4 Triangle inequality d(ij)=||vi-vj||2 is a metric
Integrality gap Ω(log log n)
SDP Solution
(1111)
(-1-1-1-1)
(111-1) (-1111) (1-111) (11-11)
(-1-1-11) (1-1-1-1) (-11-1-1) (-1-11-1)
(11-1-1) (-111-1) (-1-111) (1-1-11)
(1-11-1) (-11-11)
Formally SDP SolutionVertex equivalence class x1 x2 hellip xk (rotations)
Vector (1radick) Σj xj
Observations
bull Edge across two nodes differing in one bit
Contribution to sdp ~ 1k sdp le |E(G)|k = |E(G)|(log n)
Triangle Inequality (little bit of work and case analysis)
For most classes x1 hellip xk is ldquonearly orthogonalrdquo
Hidden Gram-Schmidt Orthogonalization Tensoring Well-separatedness
Conclusion
bull (Simple) log log n integrality gap for SCBS
bull Close the gap between log log n and log n
bull [Lee-Naor rsquo06 Cheeger-Kleiner rsquo06] Another counter-example May give (log n)c
Thank you
- Slide 1
- CS Story Sparsity
- Applications
- Estimating Sparsity
- History
- Outline of this Talk
- Slide 7
- Quadratic Program for BS
- SDP for Balanced Separator
- Why is this a Relaxation
- Quadratic Program for BS hellip
- SDP for Balanced Separator hellip
- Geometry of Triangle Ineq
- Integrality Gap Upper Bound
- Slide 15
- Math Story Metric Embeddings
- Negative Type Metrics (squared-L2)
- Embedding NEG into L1
- Cuts and L1 Metrics
- Sparsest Cut asymp Optimizing Over L1
- Metric Embeddings amp Sparsity
- Integrality Gap Lower Bound
- Slide 23
- Slide 24
- Recall Integrality Gap Lower Bound
- Slide 26
- Starting Point Hypercube()
- Hypercube hellip
- Cuts in Hypercube Coordinate
- Cuts in Hypercube hellip
- Kahn-Kalai-Linial
- Increasing Size of Balanced Cuts
- eg 4-dim hypercube
- More Formally hellip
- Slide 35
- SDP Solution
- Formally SDP Solution
- Conclusion
-
Kahn-Kalai-Linial
any balanced cut has a coordinate cut which contributes E(H) (log
k)k2 edges
Increasing Size of Balanced Cuts
consider balanced cuts in which coordinates are indistinguishable
(wrt to their contribution to the cut)
can be achieved by symmetrizing the hypercube
each coordinate contributes equally total E(H) (log k)k edges
eg 4-dim hypercube
(1111)
(-1-1-1-1)
(111-1) (-1111) (1-111) (11-11)
(-1-1-11) (1-1-1-1) (-11-1-1) (-1-11-1)
(11-1-1) (-111-1) (-1-111) (1-1-11)
(1-11-1) (-11-11)
More Formally hellipH=-11k with a rotation group acting on its coordinates
Partitions H into equivalence classes V1hellipVn
Each Vi is a vertex Edges are hypercube edges
G(VE) |E(G)|=|E(H)| k ~ log n
Balanced cuts in G correspond to balanced cuts in H
KKL any balanced cut (in H) has ldquoardquo coordinate cut which contributes E(H) (log k)k2 edges to the cut
Group is transitive ldquoeveryrdquo coordinate cut has the same contribution
Any balanced cut in G has ge E(G) (log k)k = E(G) (log log n)(log n)
Integrality Gap Lower Bound
Thm Construct graph G(1hellipnE) and unit vector assignment i -gt vi є Rn st
1 G is an ldquoexpanderrdquo every frac14 balanced cut has Ω(|E|(log log n)log n) edges (Kahn-Kalai-Linial)
2 ldquoLowrdquo SDP solution O(|E|log n)
3 Well-Separatedness Σiltj ||vi-vj||2 = n2
4 Triangle inequality d(ij)=||vi-vj||2 is a metric
Integrality gap Ω(log log n)
SDP Solution
(1111)
(-1-1-1-1)
(111-1) (-1111) (1-111) (11-11)
(-1-1-11) (1-1-1-1) (-11-1-1) (-1-11-1)
(11-1-1) (-111-1) (-1-111) (1-1-11)
(1-11-1) (-11-11)
Formally SDP SolutionVertex equivalence class x1 x2 hellip xk (rotations)
Vector (1radick) Σj xj
Observations
bull Edge across two nodes differing in one bit
Contribution to sdp ~ 1k sdp le |E(G)|k = |E(G)|(log n)
Triangle Inequality (little bit of work and case analysis)
For most classes x1 hellip xk is ldquonearly orthogonalrdquo
Hidden Gram-Schmidt Orthogonalization Tensoring Well-separatedness
Conclusion
bull (Simple) log log n integrality gap for SCBS
bull Close the gap between log log n and log n
bull [Lee-Naor rsquo06 Cheeger-Kleiner rsquo06] Another counter-example May give (log n)c
Thank you
- Slide 1
- CS Story Sparsity
- Applications
- Estimating Sparsity
- History
- Outline of this Talk
- Slide 7
- Quadratic Program for BS
- SDP for Balanced Separator
- Why is this a Relaxation
- Quadratic Program for BS hellip
- SDP for Balanced Separator hellip
- Geometry of Triangle Ineq
- Integrality Gap Upper Bound
- Slide 15
- Math Story Metric Embeddings
- Negative Type Metrics (squared-L2)
- Embedding NEG into L1
- Cuts and L1 Metrics
- Sparsest Cut asymp Optimizing Over L1
- Metric Embeddings amp Sparsity
- Integrality Gap Lower Bound
- Slide 23
- Slide 24
- Recall Integrality Gap Lower Bound
- Slide 26
- Starting Point Hypercube()
- Hypercube hellip
- Cuts in Hypercube Coordinate
- Cuts in Hypercube hellip
- Kahn-Kalai-Linial
- Increasing Size of Balanced Cuts
- eg 4-dim hypercube
- More Formally hellip
- Slide 35
- SDP Solution
- Formally SDP Solution
- Conclusion
-
Increasing Size of Balanced Cuts
consider balanced cuts in which coordinates are indistinguishable
(wrt to their contribution to the cut)
can be achieved by symmetrizing the hypercube
each coordinate contributes equally total E(H) (log k)k edges
eg 4-dim hypercube
(1111)
(-1-1-1-1)
(111-1) (-1111) (1-111) (11-11)
(-1-1-11) (1-1-1-1) (-11-1-1) (-1-11-1)
(11-1-1) (-111-1) (-1-111) (1-1-11)
(1-11-1) (-11-11)
More Formally hellipH=-11k with a rotation group acting on its coordinates
Partitions H into equivalence classes V1hellipVn
Each Vi is a vertex Edges are hypercube edges
G(VE) |E(G)|=|E(H)| k ~ log n
Balanced cuts in G correspond to balanced cuts in H
KKL any balanced cut (in H) has ldquoardquo coordinate cut which contributes E(H) (log k)k2 edges to the cut
Group is transitive ldquoeveryrdquo coordinate cut has the same contribution
Any balanced cut in G has ge E(G) (log k)k = E(G) (log log n)(log n)
Integrality Gap Lower Bound
Thm Construct graph G(1hellipnE) and unit vector assignment i -gt vi є Rn st
1 G is an ldquoexpanderrdquo every frac14 balanced cut has Ω(|E|(log log n)log n) edges (Kahn-Kalai-Linial)
2 ldquoLowrdquo SDP solution O(|E|log n)
3 Well-Separatedness Σiltj ||vi-vj||2 = n2
4 Triangle inequality d(ij)=||vi-vj||2 is a metric
Integrality gap Ω(log log n)
SDP Solution
(1111)
(-1-1-1-1)
(111-1) (-1111) (1-111) (11-11)
(-1-1-11) (1-1-1-1) (-11-1-1) (-1-11-1)
(11-1-1) (-111-1) (-1-111) (1-1-11)
(1-11-1) (-11-11)
Formally SDP SolutionVertex equivalence class x1 x2 hellip xk (rotations)
Vector (1radick) Σj xj
Observations
bull Edge across two nodes differing in one bit
Contribution to sdp ~ 1k sdp le |E(G)|k = |E(G)|(log n)
Triangle Inequality (little bit of work and case analysis)
For most classes x1 hellip xk is ldquonearly orthogonalrdquo
Hidden Gram-Schmidt Orthogonalization Tensoring Well-separatedness
Conclusion
bull (Simple) log log n integrality gap for SCBS
bull Close the gap between log log n and log n
bull [Lee-Naor rsquo06 Cheeger-Kleiner rsquo06] Another counter-example May give (log n)c
Thank you
- Slide 1
- CS Story Sparsity
- Applications
- Estimating Sparsity
- History
- Outline of this Talk
- Slide 7
- Quadratic Program for BS
- SDP for Balanced Separator
- Why is this a Relaxation
- Quadratic Program for BS hellip
- SDP for Balanced Separator hellip
- Geometry of Triangle Ineq
- Integrality Gap Upper Bound
- Slide 15
- Math Story Metric Embeddings
- Negative Type Metrics (squared-L2)
- Embedding NEG into L1
- Cuts and L1 Metrics
- Sparsest Cut asymp Optimizing Over L1
- Metric Embeddings amp Sparsity
- Integrality Gap Lower Bound
- Slide 23
- Slide 24
- Recall Integrality Gap Lower Bound
- Slide 26
- Starting Point Hypercube()
- Hypercube hellip
- Cuts in Hypercube Coordinate
- Cuts in Hypercube hellip
- Kahn-Kalai-Linial
- Increasing Size of Balanced Cuts
- eg 4-dim hypercube
- More Formally hellip
- Slide 35
- SDP Solution
- Formally SDP Solution
- Conclusion
-
eg 4-dim hypercube
(1111)
(-1-1-1-1)
(111-1) (-1111) (1-111) (11-11)
(-1-1-11) (1-1-1-1) (-11-1-1) (-1-11-1)
(11-1-1) (-111-1) (-1-111) (1-1-11)
(1-11-1) (-11-11)
More Formally hellipH=-11k with a rotation group acting on its coordinates
Partitions H into equivalence classes V1hellipVn
Each Vi is a vertex Edges are hypercube edges
G(VE) |E(G)|=|E(H)| k ~ log n
Balanced cuts in G correspond to balanced cuts in H
KKL any balanced cut (in H) has ldquoardquo coordinate cut which contributes E(H) (log k)k2 edges to the cut
Group is transitive ldquoeveryrdquo coordinate cut has the same contribution
Any balanced cut in G has ge E(G) (log k)k = E(G) (log log n)(log n)
Integrality Gap Lower Bound
Thm Construct graph G(1hellipnE) and unit vector assignment i -gt vi є Rn st
1 G is an ldquoexpanderrdquo every frac14 balanced cut has Ω(|E|(log log n)log n) edges (Kahn-Kalai-Linial)
2 ldquoLowrdquo SDP solution O(|E|log n)
3 Well-Separatedness Σiltj ||vi-vj||2 = n2
4 Triangle inequality d(ij)=||vi-vj||2 is a metric
Integrality gap Ω(log log n)
SDP Solution
(1111)
(-1-1-1-1)
(111-1) (-1111) (1-111) (11-11)
(-1-1-11) (1-1-1-1) (-11-1-1) (-1-11-1)
(11-1-1) (-111-1) (-1-111) (1-1-11)
(1-11-1) (-11-11)
Formally SDP SolutionVertex equivalence class x1 x2 hellip xk (rotations)
Vector (1radick) Σj xj
Observations
bull Edge across two nodes differing in one bit
Contribution to sdp ~ 1k sdp le |E(G)|k = |E(G)|(log n)
Triangle Inequality (little bit of work and case analysis)
For most classes x1 hellip xk is ldquonearly orthogonalrdquo
Hidden Gram-Schmidt Orthogonalization Tensoring Well-separatedness
Conclusion
bull (Simple) log log n integrality gap for SCBS
bull Close the gap between log log n and log n
bull [Lee-Naor rsquo06 Cheeger-Kleiner rsquo06] Another counter-example May give (log n)c
Thank you
- Slide 1
- CS Story Sparsity
- Applications
- Estimating Sparsity
- History
- Outline of this Talk
- Slide 7
- Quadratic Program for BS
- SDP for Balanced Separator
- Why is this a Relaxation
- Quadratic Program for BS hellip
- SDP for Balanced Separator hellip
- Geometry of Triangle Ineq
- Integrality Gap Upper Bound
- Slide 15
- Math Story Metric Embeddings
- Negative Type Metrics (squared-L2)
- Embedding NEG into L1
- Cuts and L1 Metrics
- Sparsest Cut asymp Optimizing Over L1
- Metric Embeddings amp Sparsity
- Integrality Gap Lower Bound
- Slide 23
- Slide 24
- Recall Integrality Gap Lower Bound
- Slide 26
- Starting Point Hypercube()
- Hypercube hellip
- Cuts in Hypercube Coordinate
- Cuts in Hypercube hellip
- Kahn-Kalai-Linial
- Increasing Size of Balanced Cuts
- eg 4-dim hypercube
- More Formally hellip
- Slide 35
- SDP Solution
- Formally SDP Solution
- Conclusion
-
More Formally hellipH=-11k with a rotation group acting on its coordinates
Partitions H into equivalence classes V1hellipVn
Each Vi is a vertex Edges are hypercube edges
G(VE) |E(G)|=|E(H)| k ~ log n
Balanced cuts in G correspond to balanced cuts in H
KKL any balanced cut (in H) has ldquoardquo coordinate cut which contributes E(H) (log k)k2 edges to the cut
Group is transitive ldquoeveryrdquo coordinate cut has the same contribution
Any balanced cut in G has ge E(G) (log k)k = E(G) (log log n)(log n)
Integrality Gap Lower Bound
Thm Construct graph G(1hellipnE) and unit vector assignment i -gt vi є Rn st
1 G is an ldquoexpanderrdquo every frac14 balanced cut has Ω(|E|(log log n)log n) edges (Kahn-Kalai-Linial)
2 ldquoLowrdquo SDP solution O(|E|log n)
3 Well-Separatedness Σiltj ||vi-vj||2 = n2
4 Triangle inequality d(ij)=||vi-vj||2 is a metric
Integrality gap Ω(log log n)
SDP Solution
(1111)
(-1-1-1-1)
(111-1) (-1111) (1-111) (11-11)
(-1-1-11) (1-1-1-1) (-11-1-1) (-1-11-1)
(11-1-1) (-111-1) (-1-111) (1-1-11)
(1-11-1) (-11-11)
Formally SDP SolutionVertex equivalence class x1 x2 hellip xk (rotations)
Vector (1radick) Σj xj
Observations
bull Edge across two nodes differing in one bit
Contribution to sdp ~ 1k sdp le |E(G)|k = |E(G)|(log n)
Triangle Inequality (little bit of work and case analysis)
For most classes x1 hellip xk is ldquonearly orthogonalrdquo
Hidden Gram-Schmidt Orthogonalization Tensoring Well-separatedness
Conclusion
bull (Simple) log log n integrality gap for SCBS
bull Close the gap between log log n and log n
bull [Lee-Naor rsquo06 Cheeger-Kleiner rsquo06] Another counter-example May give (log n)c
Thank you
- Slide 1
- CS Story Sparsity
- Applications
- Estimating Sparsity
- History
- Outline of this Talk
- Slide 7
- Quadratic Program for BS
- SDP for Balanced Separator
- Why is this a Relaxation
- Quadratic Program for BS hellip
- SDP for Balanced Separator hellip
- Geometry of Triangle Ineq
- Integrality Gap Upper Bound
- Slide 15
- Math Story Metric Embeddings
- Negative Type Metrics (squared-L2)
- Embedding NEG into L1
- Cuts and L1 Metrics
- Sparsest Cut asymp Optimizing Over L1
- Metric Embeddings amp Sparsity
- Integrality Gap Lower Bound
- Slide 23
- Slide 24
- Recall Integrality Gap Lower Bound
- Slide 26
- Starting Point Hypercube()
- Hypercube hellip
- Cuts in Hypercube Coordinate
- Cuts in Hypercube hellip
- Kahn-Kalai-Linial
- Increasing Size of Balanced Cuts
- eg 4-dim hypercube
- More Formally hellip
- Slide 35
- SDP Solution
- Formally SDP Solution
- Conclusion
-
Integrality Gap Lower Bound
Thm Construct graph G(1hellipnE) and unit vector assignment i -gt vi є Rn st
1 G is an ldquoexpanderrdquo every frac14 balanced cut has Ω(|E|(log log n)log n) edges (Kahn-Kalai-Linial)
2 ldquoLowrdquo SDP solution O(|E|log n)
3 Well-Separatedness Σiltj ||vi-vj||2 = n2
4 Triangle inequality d(ij)=||vi-vj||2 is a metric
Integrality gap Ω(log log n)
SDP Solution
(1111)
(-1-1-1-1)
(111-1) (-1111) (1-111) (11-11)
(-1-1-11) (1-1-1-1) (-11-1-1) (-1-11-1)
(11-1-1) (-111-1) (-1-111) (1-1-11)
(1-11-1) (-11-11)
Formally SDP SolutionVertex equivalence class x1 x2 hellip xk (rotations)
Vector (1radick) Σj xj
Observations
bull Edge across two nodes differing in one bit
Contribution to sdp ~ 1k sdp le |E(G)|k = |E(G)|(log n)
Triangle Inequality (little bit of work and case analysis)
For most classes x1 hellip xk is ldquonearly orthogonalrdquo
Hidden Gram-Schmidt Orthogonalization Tensoring Well-separatedness
Conclusion
bull (Simple) log log n integrality gap for SCBS
bull Close the gap between log log n and log n
bull [Lee-Naor rsquo06 Cheeger-Kleiner rsquo06] Another counter-example May give (log n)c
Thank you
- Slide 1
- CS Story Sparsity
- Applications
- Estimating Sparsity
- History
- Outline of this Talk
- Slide 7
- Quadratic Program for BS
- SDP for Balanced Separator
- Why is this a Relaxation
- Quadratic Program for BS hellip
- SDP for Balanced Separator hellip
- Geometry of Triangle Ineq
- Integrality Gap Upper Bound
- Slide 15
- Math Story Metric Embeddings
- Negative Type Metrics (squared-L2)
- Embedding NEG into L1
- Cuts and L1 Metrics
- Sparsest Cut asymp Optimizing Over L1
- Metric Embeddings amp Sparsity
- Integrality Gap Lower Bound
- Slide 23
- Slide 24
- Recall Integrality Gap Lower Bound
- Slide 26
- Starting Point Hypercube()
- Hypercube hellip
- Cuts in Hypercube Coordinate
- Cuts in Hypercube hellip
- Kahn-Kalai-Linial
- Increasing Size of Balanced Cuts
- eg 4-dim hypercube
- More Formally hellip
- Slide 35
- SDP Solution
- Formally SDP Solution
- Conclusion
-
SDP Solution
(1111)
(-1-1-1-1)
(111-1) (-1111) (1-111) (11-11)
(-1-1-11) (1-1-1-1) (-11-1-1) (-1-11-1)
(11-1-1) (-111-1) (-1-111) (1-1-11)
(1-11-1) (-11-11)
Formally SDP SolutionVertex equivalence class x1 x2 hellip xk (rotations)
Vector (1radick) Σj xj
Observations
bull Edge across two nodes differing in one bit
Contribution to sdp ~ 1k sdp le |E(G)|k = |E(G)|(log n)
Triangle Inequality (little bit of work and case analysis)
For most classes x1 hellip xk is ldquonearly orthogonalrdquo
Hidden Gram-Schmidt Orthogonalization Tensoring Well-separatedness
Conclusion
bull (Simple) log log n integrality gap for SCBS
bull Close the gap between log log n and log n
bull [Lee-Naor rsquo06 Cheeger-Kleiner rsquo06] Another counter-example May give (log n)c
Thank you
- Slide 1
- CS Story Sparsity
- Applications
- Estimating Sparsity
- History
- Outline of this Talk
- Slide 7
- Quadratic Program for BS
- SDP for Balanced Separator
- Why is this a Relaxation
- Quadratic Program for BS hellip
- SDP for Balanced Separator hellip
- Geometry of Triangle Ineq
- Integrality Gap Upper Bound
- Slide 15
- Math Story Metric Embeddings
- Negative Type Metrics (squared-L2)
- Embedding NEG into L1
- Cuts and L1 Metrics
- Sparsest Cut asymp Optimizing Over L1
- Metric Embeddings amp Sparsity
- Integrality Gap Lower Bound
- Slide 23
- Slide 24
- Recall Integrality Gap Lower Bound
- Slide 26
- Starting Point Hypercube()
- Hypercube hellip
- Cuts in Hypercube Coordinate
- Cuts in Hypercube hellip
- Kahn-Kalai-Linial
- Increasing Size of Balanced Cuts
- eg 4-dim hypercube
- More Formally hellip
- Slide 35
- SDP Solution
- Formally SDP Solution
- Conclusion
-
Formally SDP SolutionVertex equivalence class x1 x2 hellip xk (rotations)
Vector (1radick) Σj xj
Observations
bull Edge across two nodes differing in one bit
Contribution to sdp ~ 1k sdp le |E(G)|k = |E(G)|(log n)
Triangle Inequality (little bit of work and case analysis)
For most classes x1 hellip xk is ldquonearly orthogonalrdquo
Hidden Gram-Schmidt Orthogonalization Tensoring Well-separatedness
Conclusion
bull (Simple) log log n integrality gap for SCBS
bull Close the gap between log log n and log n
bull [Lee-Naor rsquo06 Cheeger-Kleiner rsquo06] Another counter-example May give (log n)c
Thank you
- Slide 1
- CS Story Sparsity
- Applications
- Estimating Sparsity
- History
- Outline of this Talk
- Slide 7
- Quadratic Program for BS
- SDP for Balanced Separator
- Why is this a Relaxation
- Quadratic Program for BS hellip
- SDP for Balanced Separator hellip
- Geometry of Triangle Ineq
- Integrality Gap Upper Bound
- Slide 15
- Math Story Metric Embeddings
- Negative Type Metrics (squared-L2)
- Embedding NEG into L1
- Cuts and L1 Metrics
- Sparsest Cut asymp Optimizing Over L1
- Metric Embeddings amp Sparsity
- Integrality Gap Lower Bound
- Slide 23
- Slide 24
- Recall Integrality Gap Lower Bound
- Slide 26
- Starting Point Hypercube()
- Hypercube hellip
- Cuts in Hypercube Coordinate
- Cuts in Hypercube hellip
- Kahn-Kalai-Linial
- Increasing Size of Balanced Cuts
- eg 4-dim hypercube
- More Formally hellip
- Slide 35
- SDP Solution
- Formally SDP Solution
- Conclusion
-
Conclusion
bull (Simple) log log n integrality gap for SCBS
bull Close the gap between log log n and log n
bull [Lee-Naor rsquo06 Cheeger-Kleiner rsquo06] Another counter-example May give (log n)c
Thank you
- Slide 1
- CS Story Sparsity
- Applications
- Estimating Sparsity
- History
- Outline of this Talk
- Slide 7
- Quadratic Program for BS
- SDP for Balanced Separator
- Why is this a Relaxation
- Quadratic Program for BS hellip
- SDP for Balanced Separator hellip
- Geometry of Triangle Ineq
- Integrality Gap Upper Bound
- Slide 15
- Math Story Metric Embeddings
- Negative Type Metrics (squared-L2)
- Embedding NEG into L1
- Cuts and L1 Metrics
- Sparsest Cut asymp Optimizing Over L1
- Metric Embeddings amp Sparsity
- Integrality Gap Lower Bound
- Slide 23
- Slide 24
- Recall Integrality Gap Lower Bound
- Slide 26
- Starting Point Hypercube()
- Hypercube hellip
- Cuts in Hypercube Coordinate
- Cuts in Hypercube hellip
- Kahn-Kalai-Linial
- Increasing Size of Balanced Cuts
- eg 4-dim hypercube
- More Formally hellip
- Slide 35
- SDP Solution
- Formally SDP Solution
- Conclusion
-