Random Graph Coloring

Transcription

1 Random Graph Coloring Amin Coja-Oghlan Goethe University Frankfurt

2 A simple question... What is the chromatic number of G(n,m)? [ER 60] n vertices m = dn/2 random edges

3 ... that lacks a simple answer Early work: a factor two approximation the greedy algorithm [GMcD 75; BE 76] sparse case [SU 84] Getting the asymptotics right factor 4 3 approximation [M 87] factor 1 + o(1) for d n 2/3... [B 88]... and indeed for d 1 [Ł 91] Concentration results Concentration within O( n). [SS 1987] Two-point concentration for d n 1/6... [Ł 1991]... and in fact for d n 1/2. [AK 1997]

4 Two moments do not suffice The k-colorability threshold [ER 60] consider G = G(n, m) with 2m/n d let Z k (G) = #k-colorings 1st moment d k col (2k 1)lnk 2nd moment d k col (2k 2)lnk [AN 05] improved bound d k col (2k 1)lnk 1 + o(1) [CO 13]

5 The cavity method The cavity method A generic but recipe. [ Belief/Survey Propagation ] A precise prediction as to the k-colorability threshold. A variety of predictions in mathematical physics, information theory, probabilistic combinatorics, compressive sensing.

6 The cavity method Conjectures [KMRTSZ 07] the k-colorability threshold is d k col = (2k 1)lnk 1 + o(1) there occurs a condensation phase transition at d k,cond = (2k 1)lnk 2ln2 + o(1) non-rigorous calculations based on Belief Propagation

7 Replica symmetry breaking k lnk d k,cond Replica symmetry Can walk from one coloring to any another. Only short-range effects matter. Simple coloring algorithms succeed.

8 Replica symmetry breaking k lnk d k,cond Dynamic replica symmetry breaking The set of k-colorings shatters into tiny clusters.[aco 08, M 12] Long-range effects emerge, stalling algorithms. Yet pairs of solutions look uncorrelated.

9 Replica symmetry breaking k lnk d k,cond Condensation A bounded number of clusters dominate. Pairs of solutions are heavily correlated. A second phase transition.

10 The entropy crisis dk-cond as d d k,cond, both E n Z k (G) and the cluster size drop at d k,cond they equalise

11 Chasing the k-colorability threshold Theorem [BCOHRV 13] We have d k col d k,cond. d k,cond = (2k 1)lnk 2ln2 + ε k. Within 2ln2 + o k (1) 1.39 of the first moment.

12 The condensation phase transition Theorem [BCOHRV 14] Assume k > k 0 and d > (2k 1)lnk 2. Define BP : P ([k]) γ P ([k]), BP[µ 1,...,µ γ ]( ) 1 µ h ( ) T : P 2 ([k]) P 2 ([k]), d γ [ exp( d) π γ!z γ (π) γ=0 h [k] i [γ] h [γ] Then T has a unique frozen fixed point π d,k. ] 1 µ i (h) δ BP[µ1,...,µ γ ]d π(µ j ) j [γ]

13 The condensation phase transition B(π) = B e (π) + 1 B v (π;i ;γ) ( ) d γh exp( d/(k 1)) k i [k] γ1,...,γk=0 h [k] k 1 γ h! B e d k [ (π) = ln 1 ] 2 µ 1 (h)µ 2 (h) d π hi (µ i ) 2k(k 1) h1=1 h2 [k]\{h1} h [k] i=1 [ ] k B v γ h (π;i ;γ) = ln 1 µ (j ) (h) d γ h π h h (µ (j ) ) h h=1 h [k]\{i } j =1 h [k] j =1 Theorem (ctd.) Further, d k(1 1/k) d/2 exp(b(π d,k )) [BCOHRV 14] has a unique zero d k,cond. d < d k,cond lime n Z k (G) = k(1 1/k) d/2 d > d k,cond limsupe n Z k (G) < k(1 1/k) d/2 Implies that d k col d k,cond (2k 1)lnk 2ln2

14 Random regular graphs Theorem For large k there is d k reg s.t. the random regular graph [COEH 13] is k-colorable w.h.p. if d < d k reg fails to be k-colorable w.h.p. if d > d k reg about 61% of the time d k reg is not an integer small subgraph conditioning [KPGW 10]

15 The second moment method Let Z (G) 0 and Z (G) > 0 only if G is k-colorable. Suppose 0 < E[Z 2 ] C E[Z ] 2 with C = C (k) > 0. By the Paley-Zygmund inequality, [ ] E[Z ]2 P[G is k-col] P Z > 0 E[Z 2 ] > 0. Lemma [AF 99] If liminfp[g is k-col] > 0 for some d, then d k col d o(1).

16 The Birkhoff polytope Call σ : [n] [k] balanced if σ 1 (i ) = n k for all i. Let Z k,bal = #balanced k-colorings of G. Then 1 n lne[z k,bal] lnk + d ln(1 1/k). 2 Define the k k overlap matrix ϱ(σ,τ) by ϱ i j (σ,τ) = k n σ 1 (i ) τ 1 (j ). Doubly-stochastic because σ,τ are balanced.

17 Balanced colorings Let R = {all possible overlap matrices} and Z ϱ,bal = # { (σ,τ) balanced k-colorings with overlap ϱ }. Then and thus E[Z 2 k,bal ] = E[Z ϱ,bal ] ϱ R lne[z 2 k,bal ] max ϱ R lne[z ϱ,bal]

18 Balanced colorings We have Furthermore, lne[z 2 k,bal ] max ϱ R lne[z ϱ,bal]. 1 n lne[z ϱ,bal] f (ϱ) = H(ϱ) + E(ϱ), where H(ϱ) = lnk 1 k E(ϱ) = [ d 2 ln k ϱ i j ln(ϱ i j ) [ entropy ] i, j =1 1 2 k + 1 k 2 k ϱ 2 i j i, j =1 ] [ probability ]

19 Balanced colorings As n, R is dense in the Birkhoff polytope D = { doubly-stochastic k k matrices }. Hence, 1 n lne[z 2 k,bal ] max ϱ D f (ϱ). At the barycenter ϱ = 1 k 1 we have f ( ϱ) 2 n lne[z k,bal]. E[Z 2 k,bal ] C E[Z k,bal] 2 max ϱ D f (ϱ) is attained at ϱ.

20 The singly-stochastic bound Theorem Let S = { singly-stochastic k k matrices }. [AN 05] For d d k,an = 2k lnk 2lnk 2 we have max ϱ D f (ϱ) max f (ϱ) f ( ϱ). ϱ S Proof Optimisation over a product of simplices. Going to the 6th derivative...

21 Singly vs. doubly-stochastic For d > d k,an, max ϱ S f (ϱ) is attained near ϱ half with { 1i=j if i k/2, ϱ half,i j = if i > k/2. ϱ half fails to be doubly-stochastic. 1 k

22 Singly vs. doubly-stochastic For d > d k,cond (1 + ln2) ϱ stable = (1 1/k)id + k 2 1 satisfies f (ϱ stable ) > f ( ϱ). Thus, max ϱ D f (ϱ) > f ( ϱ).

23 Clustering Assume k lnk < d < d k,cond. Clusters C 1,...,C N. max i N C i exp( Ω(n)) Z k,bal. Clusters are well-separated. Key idea Add constraints to the maximisation problem to reflect clustering.

24 Tame colorings Definition A balanced k-coloring σ is tame if its cluster C (σ) = { τ : ϱ i i (σ,τ) > 0.51 for all i = 1,...,k } has size C (σ) E[Z k,bal ], for any balanced k-coloring τ and any 1 i, j k we have ϱ i j (σ,τ) > 0.51 ϱ i j (σ,τ) 1 ln2 k k.

25 The first moment Proposition Let Z k,tame = #good k-colorings. Then for d < d k,cond, E[Z k,tame ] E[Z k,bal ]. Proof Consider the planted model. Exhibit a frozen core. Cluster size???

26 The second moment Proposition [ second moment ] For d < d k,cond, E[Z 2 k,tame ] C E[Z k,tame] 2. Call a doubly-stochastic ϱ separable if ϱ i j > 0.51 ϱ i j 1 ln2 k k for all i, j. Call ϱ s-stable if s = # { (i, j ) : ϱ i j > 0.51 }. Let D s,tame = { all s-stable separable ϱ } k 1 and D tame = D s,tame. s=0

27 The second moment We have Note that ϱ stable D tame. 1 n lne[z 2 k,tame ] max ϱ D tame f (ϱ).

28 The second moment Key insight Let d d k,cond and 0 s < k. On D s,tame the maximiser has the form 1 α if i = j s, β if i j, i, j s µ i j (s) = γ if i, j > s, ζ otherwise.

29 The cluster size The planted model choose a random map ˆσ : [n] [k] choose a random graph Ĝ given that ˆσ is a k-coloring

30 The cluster size The cluster assuming d > (1 + ε)k lnk, define { τ 1 (j ) ˆσ 1 (j ) C (Ĝ, ˆσ) = τ : min j [k] ˆσ 1 (j ) } 0.99 equivalent to other natural definitions [M 12]

31 The cluster size Lemma We have E n Z k (G) n E[Z k (G)] iff with high probability n C (Ĝ, ˆσ) n E[Z ] k(1 1/k) d/2 [COV 13] Hence, we need to calculate n C (Ĝ, ˆσ)

32 Belief Propagation v μv w μw v w Messages messages µ (t) v w ( ) P ([k]) for any adjacent pair (v, w) initialise µ (0) v w (j ) = 1{ ˆσ(v) = j }

33 Belief Propagation w v μv w μu v u The update rule define for t 0 and adjacent (v, w) µ (t+1) v w (j ) u v\w 1 µ (t) u v (j ) let µ v w ( ) = lim t µ(t) v w ( )

34 Belief Propagation The Bethe free energy define µ v (j ) u v 1 µ u v (j ), µ (v,w) (i, j ) 1{i j }µ v w (i )µ w v (j ), B(µ ) = (1 d(v))h(µ v ) + 1 H(µ (v,w) v 2 ) Physics prediction: n C (Ĝ, ˆσ) exp(b(µ )) (v,w)

35 Warning Propagation w v λ v w λ u v u Discrete messages λ (t) v w (j ) {0,1} for any (v, w), j [k] initialise λ (0) v w (j ) = 1{ ˆσ(v) = j }

36 Warning Propagation w v λ v w λ u v u Discrete updates define for t 0 and adjacent (v, w) λ (t+1) v w (j ) = min h j max u v\w λ(t) u v (h) let λ v w (j ) = lim t λ(t) v w (j )

37 Warning Propagation w v λ v w λ u v u Frozen vertices define then λ v (j ) = 1 max u v\w λ u v (j ) λ v (j ) = 0 µ v (j ) = 0

38 List coloring Lemma W.h.p. C (Ĝ, ˆσ) is the set of all colorings such that for all v, τ(v) Λ (v) = { j [k] : λ v (j ) = 1} Lemma Obtain G Ĝ by deleting all edges {v, w} such that Λ (v) Λ (w) =. Then τ is a list coloring of Ĝ iff τ is a list coloring of G.

39 A branching process v The local structure of G pick a random vertex v and consider v... ω G... including the color lists most likely acyclic

40 A branching process v A random tree set of types L = {(i,l) : i l [k]} define a suitable distribution q = (q i,l ) on L specifically, q (i,l) = (1 exp( ϱd/(k 1)))k l k exp(ϱd/(k 1)) l 1, where ϱ = (1 exp( ϱd/(k 1))) k 1

41 A branching process v A random tree set of types L = {(i,l) : i l [k]} choose the type of the root vertex from q a vertex of type (i,l) spawns Po(d q i,l ) children of type (i,l ), provided i i, l l T = resulting tree

42 A branching process v Lemma T captures the local structure of G T is finite almost surely E T Z (T ) = exp(b(π d,k )) Corollary With high probability we have n C (Ĝ, ˆσ) exp(b(π d,k ))

43 Upper bounding the k-colorability threshold Theorem We have d k col (2k 1)lnk 1 o(1) [CO 13] first moment over Warning Propagation fixed points vanilla first moment d k col (2k 1)lnk

44 Long-range vs short-range Local structure fix t > 0 and choose v randomly t (G, v) is a tree w.h.p. the local structure converges to a Galton-Watson tree

45 Long-range vs short-range Questions A random probability measure on [k] n Are there forbidden local colorings?

46 The Boltzmann distribution let S k (G) = {k-colorings of G} and Z k (G) = S k (G) define a probability measure µ k,g : [k] V (G) [0,1], σ 1{σ S k(g)} Z k (G) for U V (G) define a distribution on [k] U by µ k,g U (σ 0 ) = µ k,g { σ [k] V (G) : x U : σ(x) = σ 0 (x) } letting σ 1,σ 2,... be independet samples from µ k,g, write 1 X (σ 1,...,σ l ) = Z k (G) l σ 1,...,σ l S k (G) X (σ 1,...,σ l )

47 Correlation decay Theorem Let d < d k,cond and fix t > 0. Then [COEJ 14] lim n 1 n v [n] E µk,g t (G,v) µ k, t (G,v) = 0. The coloring induced on the depth-t neighborhood of v is asymptotically uniform.

48 Correlation decay Theorem [COEJ 14] Let d < d k,cond and fix l > 0. Then lim 1 n n l E v 1,...,v l l µ k,g {v 1,...,v l } µ k,g {vi } i=1 = 0. asymptotic l-wise independence earlier work: d < 2(k 1)ln(k 1) [MRT 11]

49 Correlation decay Theorem [COEJ 14] Let d < d k,cond and fix l, t > 0. Then lim 1 n n l E v 1,...,v l l µ k,g t (G,v 1 ) t (G,v l ) µ k, t (G,v i ) i=1 = 0. asymptotic l-wise independence and uniformity

50 Reconstruction Corollary [COEJ 14] Assume that d < d k,cond. Then non-reconstruction in G non-reconstruction in T(d,k). previously known for d < 2(k 1) ln(k 1) reconstruction threshold in T(d,k) is k lnk [MRT 11] [E 14]

51 Bicolored graphs The random replica model Generate a random graph G. Sample two k-colorings σ 1,σ 2 uniformly and independently.

52 Bicolored graphs The planted replica model Choose σ 1,σ 2 : [n] [k] uniformly and independently. G = random graph given that σ 1,σ 2 are k-colorings. Easy to analyse: local structure converges to branching process

53 Bicolored graphs Key lemma If d < d k,cond, then random replica planted replica. [COEJ 15]

54 Bicolored graphs Completing the proof study the statistics of bicolored trees in the planted model the result carries over to the random replica model the theorem follows from an averaging argument

55 Bicolored graphs Averaging replicas To show E µk,g {v1,v 2 } µ k,g {v1 } µ k,g {v2 } 0, [GM 07] use [ 1{σ(v1 E ) = c 1 }1{σ(v 2 ) = c 2 } k 2 2 ] 2 ( { } { } = E 1 σj (v 1 ) = c 1 1 σj (v 2 ) = c 2 k 2 ) j =1

56 Summary physics-inspired rigorous proofs thorough understanding for d < d k,cond techniques generalise to other problems open problem: d k col