Counting Colorings on Cubic Graphs

Transcription

1 Counting Colorings on Cubic Graphs Chihao Zhang 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)

2 Coloring For any q 3, it is NP-hard to decide whether a graph is q-colorable

3 Counting Proper Colorings Counting is harder: #P-hard on cubic graphs for any q 3[BDGJ99]

4 Approximate Counting

5 Approximate Counting It is natural to approximate the number of proper colorings

6 Approximate Counting It is natural to approximate the number of proper colorings FPRAS/FPTAS Compute Ẑ in poly(g, 1 " ) time satisfying (1 ")Ẑ apple Z(G) apple (1 + ")Ẑ

7 Approximate Counting It is natural to approximate the number of proper colorings FPRAS/FPTAS Compute Ẑ in poly(g, 1 " ) time satisfying (1 ")Ẑ apple Z(G) apple (1 + ")Ẑ Any polynomial-factor approximation algorithm can be boosted into an FPRAS/FPTAS [JVV 1986]

8 Problem Setting

9 Problem Setting FPRAS can distinguish between zero and nonzero Z(G)

10 Problem Setting FPRAS can distinguish between zero and nonzero Z(G) Assume q +1,

11 Problem Setting FPRAS can distinguish between zero and nonzero Z(G) Assume q +1, Decision: Easy

12 Problem Setting FPRAS can distinguish between zero and nonzero Z(G) Assume q +1, Decision: Easy Exact Counting: Hard

13 Problem Setting FPRAS can distinguish between zero and nonzero Z(G) Assume q +1, Decision: Easy Exact Counting: Hard Approximate Counting:?

14 Uniqueness Threshold

15 Uniqueness Threshold q = { }

16 Uniqueness Threshold q = { } q = +1 { }

17 Uniqueness Threshold q = { } q = +1 { } The Gibbs measure is unique for q +1[Jonasson 2002]

18 Two-Spin System

19 Two-Spin System In (anti-ferromagnetic) two-spin system, the uniqueness threshold corresponds to the approximability threshold

20 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 NP 6= RP, the partition function of anti-ferromagnetic two-spin system is approximable if and only if the Gibbs measure is unique.

21 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 NP 6= RP, the partition function of anti-ferromagnetic two-spin system is approximable if and only if the Gibbs measure is unique. Similar results in multi-spin system (Coloring)?

22 Counting and Sampling

23 Counting and Sampling Theorem. [JVV 1986] FPRAS FPAUS

24 Counting and Sampling Theorem. [JVV 1986] FPRAS FPAUS fully-polynomial time approximate uniform sampler

25 Counting and Sampling Theorem. [JVV 1986] FPRAS FPAUS fully-polynomial time approximate uniform sampler It is natural to use Markov chains to design sampler

26 A Markov Chain to Sample Colorings

27 A Markov Chain to Sample Colorings Colors { }

28 A Markov Chain to Sample Colorings Colors { }

29 A Markov Chain to Sample Colorings Colors { }

30 A Markov Chain to Sample Colorings Colors { }

31 A Markov Chain to Sample Colorings Colors { } (One-site) Glauber Dynamics

32 A Markov Chain to Sample Colorings Colors { } (One-site) Glauber Dynamics A uniform sampler when q +2

33 Analysis of the Mixing Time

34 Analysis of the Mixing Time Rapid mixing of the Glauber dynamics implies FPRAS for colorings

35 Analysis of the Mixing Time Rapid mixing of the Glauber dynamics implies FPRAS for colorings It is challenging to analyze the mixing time

36 Analysis of the Mixing Time Rapid mixing of the Glauber dynamics implies FPRAS for colorings It is challenging to analyze the mixing time Relax the requirement to q +

37 Analysis of the Mixing Time Rapid mixing of the Glauber dynamics implies FPRAS for colorings It is challenging to analyze the mixing time Relax the requirement to q + Ideally, =1and =2. Current best, = 11 6 [Vigoda 1999]

38 Analysis of the Mixing Time Rapid mixing of the Glauber dynamics implies FPRAS for colorings It is challenging to analyze the mixing time Relax the requirement to q + Ideally, =1and =2. Current best, = 11 6 [Vigoda 1999] On cubic graphs, q =5(= +2) [BDGJ 1999]

39 q = +1

40 q = +1 When q = +1, then chain is no longer ergodic

41 q = +1 When q = +1, then chain is no longer ergodic Colors { }

42 q = +1 When q = +1, then chain is no longer ergodic Colors { }

43 q= When q = Colors { , then chain is no longer ergodic }

44 q= When q = Colors { , then chain is no longer ergodic } Frozen!

45 q= When q = Colors { , then chain is no longer ergodic } Frozen! The method cannot achieve the uniqueness threshold

46 Recursion Based Algorithm

47 Recursion Based Algorithm G =

48 Recursion Based Algorithm G = { }

49 Recursion Based Algorithm G = { } Pr G [ = ]= P (1 Pr [ = ]) 2. (1 Pr [ = 2{,,, } ])2

50 Recursion Based Algorithm G = { } Pr G [ = ]= P (1 Pr [ = ]) 2. (1 Pr [ = 2{,,, } ])2 This recursion can be generalized to arbitrary graphs x = f d (x 11,...,x 1q ; x 21,...x 2q ;...,x d1,...,x dq )

51 Recursion Based Algorithm G = { } Computing the marginal is equivalet to Pr G [ = ]= P (1 Pr [ = ]) 2 counting colorings [JVV 1986] 2{,,, } (1 Pr [ = ])2. This recursion can be generalized to arbitrary graphs x = f d (x 11,...,x 1q ; x 21,...x 2q ;...,x d1,...,x dq )

52 Correlation Decay

53 Correlation Decay Faithfully evaluate the recursion requires (( q) n ) time

54 Correlation Decay Faithfully evaluate the recursion requires (( q) n ) time Truncate the recursion tree at level (log n)

55 Correlation Decay Faithfully evaluate the recursion requires (( q) n ) time Truncate the recursion tree at level (log n) v v

56 Correlation Decay

57 Correlation Decay q = +1 { }

58 Correlation Decay q = +1 { } Uniqueness Condition: the correlation between the root and boundary decays.

59 Correlation Decay q = +1 { } Uniqueness Condition: the correlation between the root and boundary decays. Uniqueness threshold suggests q +1is sufficient for CD to hold

60 Sufficient Condition

61 Sufficient Condition Very hard to rigorously establish the decay property

62 Sufficient Condition Very hard to rigorously establish the decay property A sufficient condition is for some < 1 f d (x) f d ( x) apple kx xk 1

63 Sufficient Condition Very hard to rigorously establish the decay property A sufficient condition is for some < 1 f d (x) f d ( x) apple kx xk 1 Based on the idea, the best bounds so far is q> [LY 2013]

64 One Step Contraction

65 One Step Contraction Establish apple krf d (x)k 1 < 1 for x =(x 1,...,x d ) 2 D The domain D is all the possible values of marginals.

66 One Step Contraction Establish apple krf d (x)k 1 < 1 for x =(x 1,...,x d ) 2 D The domain D is all the possible values of marginals. D is too complicated to work with Introduce an upper bound u i for each x i Work with the polytope P := {x 0 apple x i apple u i }.

67 Barrier

68 Barrier The relaxation does not change the result in two-spin system [SSY 2012; LLY 2012; LLY 2013]

69 Barrier The relaxation does not change the result in two-spin system [SSY 2012; LLY 2012; LLY 2013] This is no longer true for coloring

70 Barrier The relaxation does not change the result in two-spin system [SSY 2012; LLY 2012; LLY 2013] This is no longer true for coloring It can be shown that for some > 0, the one-step contraction does not hold for q =(2+ )

71 Barrier The relaxation does not change the result in two-spin system [SSY 2012; LLY 2012; LLY 2013] This is no longer true for coloring It can be shown that for some > 0, the one-step contraction does not hold for q =(2+ ) To improve, one needs to find a better relaxation

72 General Recursion

73 General Recursion Pr G,L [c(v) =i] = Q d k=1 1 Pr G v,l [c(v k,i k)=i] P Q d j2l(v) k=1 1 Pr G v,l [c(v k,j k)=j]

74 General Recursion Pr G,L [c(v) =i] = Q d k=1 1 Pr G v,l [c(v k,i k)=i] P Q d j2l(v) k=1 1 Pr G v,l [c(v k,j k)=j] A vertex of degree d branches into d q subinstances

75 General Recursion Pr G,L [c(v) =i] = Q d k=1 1 Pr G v,l [c(v k,i k)=i] P Q d j2l(v) k=1 1 Pr G v,l [c(v k,j k)=j] A vertex of degree d branches into d q subinstances The subinstances corresponding to the first child are marginals on the same graph

76 Constraints

77 Constraints x = f d (x 11,...,x 1q ; x 21,...x 2q ;...,x d1,...,x dq )

78 Constraints x = f d (x 11,...,x 1q ; x 21,...x 2q ;...,x d1,...,x dq ) Analyze rf d on the hyperplane P q i=1 x 1i =1

79 Constraints x = f d (x 11,...,x 1q ; x 21,...x 2q ;...,x d1,...,x dq ) Analyze rf d on the hyperplane P q i=1 x 1i =1 Introduce the polytope P 0 := P \ {x P q i=1 x 1i =1}

80 Technical Issue

81 Technical Issue In fact, we need to analyze an amortized function ' f d

82 Technical Issue In fact, we need to analyze an amortized function ' f d The constraint P q i=1 x 1i =1breaks down in the new domain

83 Technical Issue In fact, we need to analyze an amortized function ' f d The constraint P q i=1 x 1i =1breaks down in the new domain The use of potential function and domain restriction are not compatible

84 New Recursion

85 New Recursion We have to work with the same polytope P

86 New Recursion We have to work with the same polytope P Pr G,L [c(v) =i] = Q d k=1 1 Pr G v,l [c(v k,i k)=i] P Q d j2l(v) k=1 1 Pr G v,l [c(v k,j k)=j] with P j2[q] Pr G v,l 1,j [c(v 1 )=j] =1

87 New Recursion We have to work with the same polytope P Pr G,L [c(v) =i] = Q d k=1 1 Pr G v,l [c(v k,i k)=i] P Q d j2l(v) k=1 1 Pr G v,l [c(v k,j k)=j] with P j2[q] Pr G v,l 1,j [c(v 1 )=j] =1 P The constraint q is implicitly imposed if i=1 x 1i =1 one further expand the first child with the same recursion

88

89 New definition of one step recursion

90 New definition of one step recursion The new recursion keeps more information

91 Cubic Graphs

92 Cubic Graphs The new constraint is negligible when the degree is large

93 Cubic Graphs The new constraint is negligible when the degree is large It turns to be useful when =3

94 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

95 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 We are able to establish CD property with the help of computer

96 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 We are able to establish CD property with the help of computer Theorem. There exists an FPTAS to compute the number of proper four-colorings on graphs with maximum degree three.

97 Final Remark

98 Final Remark The first algorithm to achieve the conjectured optimal bound in multi-spin system

99 Final Remark The first algorithm to achieve the conjectured optimal bound in multi-spin system It is possible to obtain FPTAS when q =5and =4

100 Final Remark The first algorithm to achieve the conjectured optimal bound in multi-spin system It is possible to obtain FPTAS when q =5and =4 More constraints?

101 Final Remark The first algorithm to achieve the conjectured optimal bound in multi-spin system It is possible to obtain FPTAS when q =5and =4 More constraints? Other ways to establish correlation decay