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Counting Colorings on Cubic Graphs
The Chinese University of Hong Kong
Joint work with Pinyan Lu (Shanghai U of Finance and Economics) Kuan Yang (Oxford University)
Minshen Zhu (Purdue University)
Chihao Zhang
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Coloring
For any q � 3, it is NP-hard to decidewhether a graph is q-colorable
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Counting Proper Colorings
• Counting is harder:
…
#P-hard on cubic graphs for any q � 3[BDGJ99]
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Approximate Counting
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Approximate Counting
It is natural to approximate the number of proper colorings
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Approximate Counting
It is natural to approximate the number of proper colorings
(1� ")Z Z(G) (1 + ")Z
Compute
ˆZ in poly(G, 1" ) time satisfying
FPRAS/FPTAS
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Approximate Counting
It is natural to approximate the number of proper colorings
(1� ")Z Z(G) (1 + ")Z
Compute
ˆZ in poly(G, 1" ) time satisfying
FPRAS/FPTAS
Any polynomial-factor approximation algorithm can be boosted into an FPRAS/FPTAS [JVV 1986]
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Problem Setting
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Problem Setting
FPRAS can distinguish between zero and nonzeroZ(G)
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Problem Setting
Assume q � �+ 1,
FPRAS can distinguish between zero and nonzeroZ(G)
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Problem Setting
Assume q � �+ 1,
FPRAS can distinguish between zero and nonzeroZ(G)
Decision: Easy
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Problem Setting
Assume q � �+ 1,
FPRAS can distinguish between zero and nonzeroZ(G)
Decision: Easy
Exact Counting: Hard
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Problem Setting
Assume q � �+ 1,
FPRAS can distinguish between zero and nonzeroZ(G)
Decision: Easy
Exact Counting: Hard
Approximate Counting: ?
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Uniqueness Threshold
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Uniqueness Threshold
q = �
{ }
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Uniqueness Threshold
q = �
{ }
q = �+ 1
{ }
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Uniqueness Threshold
q = �
{ }
q = �+ 1
{ }
The Gibbs measure is unique for q � �+ 1 [Jonasson 2002]
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Two-Spin System
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Two-Spin System
In (anti-ferromagnetic) two-spin system, the uniqueness threshold corresponds to the approximability threshold
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Two-Spin System
In (anti-ferromagnetic) two-spin system, the uniqueness threshold corresponds to the approximability threshold
Theorem. [Weitz06, Sly10, SS11, SST12, LLY12, LLY13] Assume , the partition function of anti-ferromagnetic two-spin system is approximable if and only if the Gibbs measure is unique.
NP 6= RP
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Two-Spin System
In (anti-ferromagnetic) two-spin system, the uniqueness threshold corresponds to the approximability threshold
Theorem. [Weitz06, Sly10, SS11, SST12, LLY12, LLY13] Assume , the partition function of anti-ferromagnetic two-spin system is approximable if and only if the Gibbs measure is unique.
NP 6= RP
Similar results in multi-spin system (Coloring) ?
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Counting and Sampling
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Counting and Sampling
FPRAS
Theorem. [JVV 1986]
FPAUS
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Counting and Sampling
FPRAS
Theorem. [JVV 1986]
FPAUS
fully-polynomial time approximate uniform sampler
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Counting and Sampling
FPRAS
Theorem. [JVV 1986]
FPAUS
fully-polynomial time approximate uniform sampler
It is natural to use Markov chains to design sampler
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A Markov Chain to Sample Colorings
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A Markov Chain to Sample Colorings
Colors { }
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A Markov Chain to Sample Colorings
Colors { }
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A Markov Chain to Sample Colorings
Colors { }
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A Markov Chain to Sample Colorings
Colors { }
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A Markov Chain to Sample Colorings
Colors { }
(One-site) Glauber Dynamics
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A Markov Chain to Sample Colorings
Colors { }
(One-site) Glauber Dynamics
A uniform sampler when q � �+ 2
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Analysis of the Mixing Time
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Analysis of the Mixing TimeRapid mixing of the Glauber dynamics implies FPRAS for colorings
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Analysis of the Mixing TimeRapid mixing of the Glauber dynamics implies FPRAS for colorings
It is challenging to analyze the mixing time
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Analysis of the Mixing TimeRapid mixing of the Glauber dynamics implies FPRAS for colorings
It is challenging to analyze the mixing time
Relax the requirement to q � ↵ ·�+ �
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Analysis of the Mixing TimeRapid mixing of the Glauber dynamics implies FPRAS for colorings
It is challenging to analyze the mixing time
Relax the requirement to q � ↵ ·�+ �
Ideally, ↵ = 1 and � = 2. Current best, ↵ = 116 [Vigoda 1999]
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Analysis of the Mixing TimeRapid mixing of the Glauber dynamics implies FPRAS for colorings
It is challenging to analyze the mixing time
Relax the requirement to q � ↵ ·�+ �
Ideally, ↵ = 1 and � = 2. Current best, ↵ = 116 [Vigoda 1999]
On cubic graphs, q = 5 (= �+ 2) [BDGJ 1999]
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q = �+ 1
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q = �+ 1
When q = �+ 1, then chain is no longer ergodic
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q = �+ 1
When q = �+ 1, then chain is no longer ergodic
Colors { }
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q = �+ 1
When q = �+ 1, then chain is no longer ergodic
Colors { }
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q = �+ 1
When q = �+ 1, then chain is no longer ergodic
Colors { }
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q = �+ 1
When q = �+ 1, then chain is no longer ergodic
Colors { }
Frozen!
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q = �+ 1
When q = �+ 1, then chain is no longer ergodic
The method cannot achieve the uniqueness threshold
Colors { }
Frozen!
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Recursion Based Algorithm
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Recursion Based Algorithm
G =
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Recursion Based Algorithm
{ }G =
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Recursion Based Algorithm
{ }G =
PrG [ = ] =(1�Pr [ = ])2
P2{ , , , }(1�Pr [ = ])2
.
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Recursion Based Algorithm
{ }G =
PrG [ = ] =(1�Pr [ = ])2
P2{ , , , }(1�Pr [ = ])2
.
This recursion can be generalized to arbitrary graphs
x = fd(x11, . . . , x1q;x21, . . . x2q; . . . , xd1, . . . , xdq)
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Recursion Based Algorithm
{ }G =
PrG [ = ] =(1�Pr [ = ])2
P2{ , , , }(1�Pr [ = ])2
.
This recursion can be generalized to arbitrary graphs
x = fd(x11, . . . , x1q;x21, . . . x2q; . . . , xd1, . . . , xdq)
Computing the marginal is equivalet to counting colorings [JVV 1986]
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Correlation Decay
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Correlation Decay
Faithfully evaluate the recursion requires ⌦ ((�q)n) time
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Correlation Decay
Faithfully evaluate the recursion requires ⌦ ((�q)n) time
Truncate the recursion tree at level ⇥(log n)
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Correlation Decay
Faithfully evaluate the recursion requires ⌦ ((�q)n) time
v v
Truncate the recursion tree at level ⇥(log n)
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Correlation Decay
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Correlation Decay
q = �+ 1
{ }
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Correlation Decay
q = �+ 1
{ }
Uniqueness Condition: the correlation between the root and boundary decays.
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Correlation Decay
Uniqueness threshold suggests q � �+ 1 is sufficient for CD to hold
q = �+ 1
{ }
Uniqueness Condition: the correlation between the root and boundary decays.
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Sufficient Condition
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Sufficient Condition
Very hard to rigorously establish the decay property
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Sufficient Condition
Very hard to rigorously establish the decay property
A sufficient condition is
|fd(x)� fd(x)| � · kx� xk1
for some � < 1
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Sufficient Condition
Very hard to rigorously establish the decay property
A sufficient condition is
|fd(x)� fd(x)| � · kx� xk1
for some � < 1
Based on the idea, the best bounds so far isq > 2.581�+ 1 [LY 2013]
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One Step Contraction
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One Step Contraction
The domain D is all the possible values of marginals.
Establish � krfd(x)k1 < 1 for x = (x1, . . . , xd) 2 DEstablish � krfd(x)k1 < 1 for x = (x1, . . . , xd) 2 D
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One Step Contraction
D is too complicated to work with
Introduce an upper bound ui for each xi
Work with the polytope P := {x | 0 xi ui}.
The domain D is all the possible values of marginals.
Establish � krfd(x)k1 < 1 for x = (x1, . . . , xd) 2 DEstablish � krfd(x)k1 < 1 for x = (x1, . . . , xd) 2 D
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Barrier
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BarrierThe relaxation does not change the result in two-spin system [SSY 2012; LLY 2012; LLY 2013]
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Barrier
This is no longer true for coloring
The relaxation does not change the result in two-spin system [SSY 2012; LLY 2012; LLY 2013]
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Barrier
It can be shown that for some , the one-step contraction does not hold for
⇠ > 0q = (2 + ⇠)�
This is no longer true for coloring
The relaxation does not change the result in two-spin system [SSY 2012; LLY 2012; LLY 2013]
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Barrier
It can be shown that for some , the one-step contraction does not hold for
⇠ > 0q = (2 + ⇠)�
To improve, one needs to find a better relaxation
This is no longer true for coloring
The relaxation does not change the result in two-spin system [SSY 2012; LLY 2012; LLY 2013]
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General Recursion
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General Recursion
PrG,L [c(v) = i] =
Qdk=1
�1�PrGv,Lk,i [c(vk) = i]
�P
j2L(v)
Qdk=1
�1�PrGv,Lk,j [c(vk) = j]
�
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General Recursion
PrG,L [c(v) = i] =
Qdk=1
�1�PrGv,Lk,i [c(vk) = i]
�P
j2L(v)
Qdk=1
�1�PrGv,Lk,j [c(vk) = j]
�
A vertex of degree d branches into d⇥ q subinstances
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General Recursion
PrG,L [c(v) = i] =
Qdk=1
�1�PrGv,Lk,i [c(vk) = i]
�P
j2L(v)
Qdk=1
�1�PrGv,Lk,j [c(vk) = j]
�
The subinstances corresponding to the first child are marginals on the same graph
A vertex of degree d branches into d⇥ q subinstances
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Constraints
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Constraints
x = fd(x11, . . . , x1q;x21, . . . x2q; . . . , xd1, . . . , xdq)
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Constraints
x = fd(x11, . . . , x1q;x21, . . . x2q; . . . , xd1, . . . , xdq)
Analyze rfd on the hyperplane
Pqi=1 x1i = 1
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Constraints
x = fd(x11, . . . , x1q;x21, . . . x2q; . . . , xd1, . . . , xdq)
Analyze rfd on the hyperplane
Pqi=1 x1i = 1
Introduce the polytope P
0 := P \ {x |Pq
i=1 x1i = 1}
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Technical Issue
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Technical Issue
In fact, we need to analyze an amortized function ' � fd
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Technical Issue
In fact, we need to analyze an amortized function ' � fd
The constraint
Pqi=1 x1i = 1 breaks down in the new domain
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Technical Issue
In fact, we need to analyze an amortized function ' � fd
The constraint
Pqi=1 x1i = 1 breaks down in the new domain
The use of potential function and domain restriction are not compatible
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New Recursion
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New Recursion
We have to work with the same polytope P
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New Recursion
We have to work with the same polytope P
PrG,L [c(v) = i] =
Qdk=1
�1�PrGv,Lk,i [c(vk) = i]
�P
j2L(v)
Qdk=1
�1�PrGv,Lk,j [c(vk) = j]
�
withP
j2[q] PrGv,L1,j [c(v1) = j] = 1
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New Recursion
We have to work with the same polytope P
PrG,L [c(v) = i] =
Qdk=1
�1�PrGv,Lk,i [c(vk) = i]
�P
j2L(v)
Qdk=1
�1�PrGv,Lk,j [c(vk) = j]
�
withP
j2[q] PrGv,L1,j [c(v1) = j] = 1
The constraint is implicitly imposed if one further expand the first child with the same recursion
Pqi=1 x1i = 1
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New definition of one step recursion
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New definition of one step recursion
The new recursion keeps more information
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Cubic Graphs
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Cubic Graphs
The new constraint is negligible when the degree is large
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Cubic Graphs
The new constraint is negligible when the degree is large
It turns to be useful when � = 3
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Cubic Graphs
The new constraint is negligible when the degree is large
It turns to be useful when � = 3
The amortized analysis is very complicated even � = 3
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Cubic Graphs
The new constraint is negligible when the degree is large
It turns to be useful when � = 3
We are able to establish CD property with the help of computer
The amortized analysis is very complicated even � = 3
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Cubic Graphs
The new constraint is negligible when the degree is large
It turns to be useful when � = 3
Theorem. There exists an FPTAS to compute the number of proper four-colorings on graphs with maximum degree three.
We are able to establish CD property with the help of computer
The amortized analysis is very complicated even � = 3
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Final Remark
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Final Remark
The first algorithm to achieve the conjectured optimal bound in multi-spin system
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Final Remark
It is possible to obtain FPTAS when q = 5 and � = 4
The first algorithm to achieve the conjectured optimal bound in multi-spin system
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Final Remark
More constraints?
It is possible to obtain FPTAS when q = 5 and � = 4
The first algorithm to achieve the conjectured optimal bound in multi-spin system
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Final Remark
More constraints?
It is possible to obtain FPTAS when q = 5 and � = 4
Other ways to establish correlation decay
The first algorithm to achieve the conjectured optimal bound in multi-spin system