vertex sparsifiers: new results from old techniques (and some open questions) robert krauthgamer...

Vertex sparsifiers: New results from old techniques (and some open questions)

Robert Krauthgamer (Weizmann Institute)

Joint work with Matthias Englert, Anupam Gupta,

Harald Räcke, Inbal Talgam-Cohen and Kunal Talwar.

Presented: Newton Institute, Jan. 2011 (w/minor corrections)


2

Graph BisectionInput: Graph G=(V,E)

Goal: partition the vertex set into V1,V2

with |V1|=|V2|,

so as to minimize e(V1,V2).

(may allow edge-capacities)

Central problem, well-studied, NP-hard …

Polynomial-time algorithm [Räcke’08]:

O(log n) approximation


3

Terminal (or Steiner) BisectionInput: Graph G=(V,E) and terminals KµV

Goal: partition the vertex set into V1,V2

with |V1ÅK|=| V2ÅK |,

so as to minimize e(V1,V2).

Same O(log n) approximation [Räcke’08].

But can we do f(k) where k=|K|?

Similarly, Steiner versions of Linear Arrangement, Oblivious Routing, etc.


4

Vertex Sparsifiers (w.r.t. Cuts)Input: Graph G=(V,E) and terminals KµV

Goal: A graph H on vertex set K, such that

for every partition K=S[T,

MinCutG(S,T) ¼ MinCutH(S,T).

(we allow edge-capacities)

Why “compress” graph G “onto” terminal set K? Information-theory: Efficiently represent 2k values Computation: Reduce problem size/approximation

Stronger version: preserve all multi-commodity flows among terminals K.

5 94

8

35

2

1

9

8

G

4

31

H


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Vertex Sparsifiers – Previous work [Moitra’09, Leighton-Moitra’10]

There are (flow) sparsifiers with quality .

Can efficiently find one with quality .

Yields approximation for Terminal Bisection

Similarly, approximation for other problems

But H is not “simple” Even if G is

O( logklog logk )

O( log2 klog logk )

O( log3 klog logk )

polylog(k)


6

Our Results Can efficiently find a sparsifier with quality .

Can efficiently find a tree-based sparsifier with quality .

Yields approximation for Terminal Bisection.

Similar improvements for other problems

If G is planar, then quality is O(1) and H is planar-based In fact, only use minors of G Holds for every minor-closed family

O( logklog logk )

O(logk)

O(logk)

Convex combination of “dominating” trees

Similar results (and lower bounds) were proved independently by Makarychev-Makarychev and by Charikar-Leighton-Li-Moitra.


7


8

Flow–Distance DualityConnection between: Sparsifier: faithful representation of flows Embedding: faithful representation of distances

Transfer Theorem [Räcke’08, Andersen-Feige’09].

Fix a graph G and a collection M of mappings M:EP(E). Then: For all edge-lengths l:ER+ there is a probabilistic mapping with

stretch (distortion) ½¸1 m

For all edge-capacities c:ER+ there is a probabilistic mapping with quality (congestion) ½¸1

Moreover, there is efficient algorithm for one iff for the other.

convex combination of mappings


9

Edge Mappings Fix G=(V,E), and let P(E) be all multisets of E (typically paths). A mapping M:EP(E) can be represented as a matrix M in ZE£E

where Me,f = number of occurrences of f in M(e).

Illustration: Embed V to a dominating tree T=(V,ET)

For xy2ET fix x-y path in G (e.g. shortest) Let M(uv2E) = {“map” u-v path in T into G}.

But how to choose M?

G1

n23

s13

e

M(e)s1n

2 3 n

1T

…


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0-extensions Defn: A 0-extension of (G=(V,E), lG) with terminals KµV to be

a retraction f:V K; along with a graph (H=(K,EH),lH) where lH(x,y)=dG(x,y) for all (x,y)2 EH.

) dH dominates dG [on pairs in K]

G1

n23 2 3 n

1H

Defn: Stretch of a probabilistic 0-extension is the minimum ®¸1 s.t.

EH[dH(f(x),f(y))] · ® dG(x,y) for all x,y2V

3010

20


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Tree 0-extensions Example 1: Graph H is a tree call it a tree 0-extension

Corollary of [Gupta-Nagarajan-Ravi’10]: There is an algorithm that produces tree 0-extensions with stretch ®=O(log k)

Idea: Use variant of [Fakcharoenphol-Rao-Talwar’04] but Allow distance between non-terminals to contract “Remap” non-terminals leaves to terminals “Purge” internal (Steiner) nodes [Gupta’01]

Now use the Transfer Theorem: M = all tree 0-extensions Distance mappings exist with stretch O(log k) Thus get a tree-based sparsifier with quality O(log k)

G1

n23 2 3 n

1H

3010

20


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Induced 0-extensions Example 2: Graph H is “induced by G” via EH={ (f(u),f(v)) : (u,v)2 E }

call this H=Hf an induced 0-extension.

Theorem [Fakcharoenphol-Harrelson-Rao-Talwar’04]: There is an algorithm producing induced 0-extensions with ®=O(log k / loglog k)

Now use the Transfer Theorem: M = all induced 0-extensions Distance mappings exist with stretch O(log k / loglog k) Thus get a sparsifier with quality O(log k / loglog k)

G1

n23

10

20

2 3 n

1Hf


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Planar Graphs Theorem [Calinescu-Karloff-Rabani’04]: There is an algorithm

producing induced 0-extensions with ®=O(1)

Now use the Transfer Theorem: M = all induced 0-extensions Distance mappings exist with stretch O(1) Thus get a sparsifier with quality O(1)

Idea: Make sure Hf is a minor of G. Hence planarity is guaranteed.

We would like the sparsifier to be planar!!


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Connected 0-extension Defn. A 0-extension f:VK is called connected if each f-1(x) induces

a connected subgraph of G.

Observe: f is connected ) Hf is a minor of G ) Hf is planar

We give first algorithms for connected 0-extension: For planar graphs: we achieve stretch O(1) For ¯-decomposable metrics: stretch O(¯ log ¯) For general metrics: stretch O(log k)

Via Transfer Theorem: planar-based sparsifier with quality O(1) etc.

Not connected:1

n23

10

20Connected:

1

n23

10

20


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Implications to Metric Embedding Theorem [Gupta’01]: For every tree T and terminals K, there is a

tree on K that represents all distances faithfully (factor 8)

2 3 n

1T

…

This work: For every planar graph G and terminals K, there is a (probabilistic) planar graph on K that represents all distances faithfully (expected O(1) stretch)

Simplifies embedding results

3 4 n

2T’

…

22 2


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Connected 0-extension in Planar MetricsAlgorithm (Input: Graph G with edge-lengths l and terminals K)1. Init: f(v)=v for v2K and f(v)=? for v2VnK.

2. For each r=1,2,…,2i,…,diam(V)

3. sample ¯-decomposition P of dG with diameter r

4. for each C’2P containing both mapped and unmapped vertices

5. delete from C’ mapped vertices

6. for each connected component C in C’

7. choose vertex wC2C’ that was deleted and has edge to C

8. reset f(u)=f(wC) for all u2C G

Connectivity: by construction Diameter: at time r, vertices are

mapped to terminals within O(r) Stretch: Prob. to settle (u,v) at “late”

time r is 1/r2 (must be separate twiced)

Pr[P(x)P(y)] · ¯ dG(x,y) / r


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Open Problems Steiner Points Removal: Given planar graph G and terminals K,

build a single planar graph only on K that represents all distances faithfully Apparently possible for outerplanar graphs [Basu-Gupta’08] More generally: same for general G, using minors

s-sparse extension: Given a graph G and terminals K, choose S¶K of size s, and a 0-extension (retraction) into this S Is there a poly(k)-sparse extension of expected stretch O(1)?

Is there a single (non-probabilistic) planar sparsifier graph? More generally: extend duality between Distances and Capacities,

perhaps to level of a single graph, or to “preserve” minors

Analogous questions for cuts (e.g. SPR, few “pseudo-terminals”) Analogous questions for Euclidean metrics (e.g. what is “minor”)

vertex sparsifiers: new results from old techniques (and some open questions) robert krauthgamer...

Documents

h slide

new results

v goal

old techniques

graph h

extension of g

e goal

e h d h fx