1. Why almost all k-colorable graphs are easy A. Coja-Oghlan, M. Krivelevich, D. Vilenchik

2. Talk Outline Random graphs: phase transitions and clustering How do typical k -colorable graphs look? Message passing and clustering (SAT) Proof techniques – two approaches

3. The k -Coloring Problem Given a graph G =( V, E ): Find f : V ! [ k ] s.t. 8 ( u, v ) 2 E ( G ): f ( u )  f ( v ) Find f with minimal possible k Such k is called the chromatic number of G,  ( G ) E.g.  ( G )=3 3 4 12

4. The k -Coloring Problem Finding a proper k -coloring is NP Hard No polynomial time algorithm approximates  ( G ) within factor better than n 1 -  (unless NP µ ZPP) [FK98] How to proceed? random models and average case analysis G n, p - every possible edge is included w.p. p = p ( n )  ( G n, p )= np / 2ln ( np ) for np 2 [ c 0, n / log 7 n ] [Bol88,Luc91]

5. Phase transitions and clustering Consider the variant G n, m of G n, p : Choose uniformly at random m = m ( n ) edges When, G n, m and G n, p are “close” There exists a constant d = d ( k ) such that 2m / n > d : almost all graphs in G n, m are not k -colorable 2m / n < d : almost all graphs are k -colorable [Fri99] Such phenomena is called a phase transition

6. Phase transitions and clustering G n, m with 2m / n just below the threshold is “hard” experimentally Possible explanation (partially non-rigorous) comes from statistical physics [MPWZ02] The “geometrical” structure of the space of proper k -colorings - the clustering phenomena Need to define notion of distance

7. Phase transitions and clustering Two k -colorings are the same if they differ only by a permutation of the color classes Two k -colorings ,  are at distance t if they disagree on the color of at least t vertices in every permutation of the color classes. There exists one permutation obtaining equality Similar to Hamming distance

8. Phase transitions and clustering G n, m with 2m / n just below the threshold: All colorings within a cluster are “close” A linear number of vertices are “frozen” Every two clusters are “far” from each other Exponentially many clusters based on analysis that uses partially- rigorous tools Proved rigorously for k -SAT, k ¸ 8 [AR06,MMZ05] For k -SAT: not believed to be true for small k, say k = 3 [MMW05]

9. Phase transitions and clustering Why does this structure make life hard? Heuristics get “distracted” by this structure Every cluster “pulls” in its direction Heuristics try to find a compromise between clusters This is impossible due to the structure Survey Propagation does well in practice [BMWZ05]

10. Random k-colorable graphs G n, m with 2m / n above the threshold – not suitable to study k -colorable graphs Instead, consider G n, m | { k -colorability } The uniform distribution over k -colorable graphs with exactly m edges Another possibility, the planted model G n, m, k Partition the vertex set into k color classes of size n / k Include m random edges that respect the coloring V1V1 V3V3 V2V2

11. Our Results Characterization of G n, m | { k –colorability } 2m / n = C k, C k a sufficiently large constant Using rigorous analysis we show that typically: Single cluster of proper k -colorings Size of the cluster is exponential in n (1-exp{-  ( C k )})n vertices are “frozen”

12. Our Results There exists a deterministic polynomial time algorithm that k -colors almost all k -colorable graphs with m > C k n edges. C k a sufficiently large constant. Rigorously complement results for sparse case: When clustering is simple – the problem is easy When clustering is “complicated” – the problem is harder (?) Almost all k -colorable graphs are easy !

13. Our Results Show that G n, m, k and G n, m | { k –colorability } share many structural properties (“close”) Justifying the somewhat unnatural usage of planted- solution models Alon-Kahale’s coloring algorithm [AK97] works for G n, m | { k –colorability } as well G n, m, k also has the same clustering structure

14. Our Results Our results also apply to the k -SAT setting Similar threshold and clustering phenomena are known/believed for k -SAT The planted and uniform SAT distributions are “close” Flaxman’s algorithm for planted 3CNF formulas works for the uniform setting Improving the exponential time algorithm for uniform satisfiable 3CNFs (only one known so far) Answering open research questions in [BBG02]

15. What was known so far? UniformPlanted Dist. Density Clustering phenomena Survey Propagation < d k Clustering Planted and Uniform “close” Alon and Kahale’s coloring algorithm [AK97] Alon and Kahale’s coloring algorithm [AK97] CkCk Alon-Kahale’s coloring algorithm [AK97] Planted and Uniform coincide [AK97] C k log n

16. What was known for SAT? UniformPlanted Dist. Density Clustering phenomena Survey Propagation [BMZ05] > d k Exponential time algorithm [Chen03] Planted and Uniform “close” Flaxman, k-opt Clustering Flaxman’s algorithm [Fla03] Version of k-opt [FV04] CkCk Majority vote works whp [BBG02] Planted and Uniform coincide Majority vote C k log n

17. Algorithmic Perspective Show that Alon and Kahale’s algorithm [AK97] works in the uniform case What is Alon and Kahale’s algorithm? Approximate a proper 3-coloring (spectral techniques) Refine the coloring – recoloring step Uncolor “suspicious” vertices G [ U ] – graph induced by uncolored vertices Exhaustively color G [ U ] according to G [ V \ U ] Outcome differs from planted on n /1000 vertices Outcome agrees on the core Core remains colored Every colored vertex agrees with planted Logarithmic size connected components

18. Algorithmic Perspective - SAT Show that Flaxman’s algorithm [Fla03] works in the uniform case What is Flaxman’s algorithm? Approximate a satisfying assignment (majority vote) Unassign “suspicious” variables G [ U ] – graph induced by unassigned variables Exhaustively satisfy G [ U ] according to G [ V \ U ]

19. SAT and Message Passing Warning Propagation: Given a 3CNF F – define Factor Graph G ( F ) Bipartite graph: V 1 = variables, V 2 = clauses ( x, C ) 2 E ( G ) iff x appears in C Two types of messages: C =( x Ç y Ç z ) C  x = 1 if y  C < 0 and z  C < 0; 0 otherwise x  C = (  x 2 C ’, C ’  C C ’  x ) – (  ¬ x 2 C ’’  C ’’  x )

20. SAT and Message Passing WP( F ) Initialize all messages C  x to 1/0 w.p. 0.5 Repeat until no message changes: Randomly order the edges of G ( F ) Evaluate all messages C  x Assign every x according to (  x 2 C ’ C ’  x ) – (  ¬ x 2 C ’’ C ’’  x ) Theorem [FMV06]: If F sampled according to Planted 3SAT p = d / n 2, d sufficiently large constant, then whp: WP converges after O( logn ) iterations Assigned variables agree with some satisfying assignment All but exp{-  ( d )} n variables are assigned Clauses of unassigned variables are “easy” to satisfy

21. SAT and Message Passing Our work implies – [FMV06] applies for the uniform SAT setting as well Reinforces the following thesis: When clustering is complicated ) formulas are hard ) sophisticated algorithms needed: Survey Propagation When clustering is simple ) formulas are easy ) naïve algorithms work: Warning Propagation

23. Clustering: Proof Techniques Recall, G n, m | { k -colorability } The uniform distribution over k -colorable graphs with exactly m edges Why more difficult than the planted distribution? Edges are not independent For starters, consider the planted distribution G n, p, k ( k = 3 ) V1V1 V3V3 V2V2

24. Proof Techniques – The Core Every vertex is expected to have d / 3 neighbors in every other color class ( d = np ) Claim 1: whp there is no subgraph H of G s.t. | V ( H )| d | H |/ 10 Claim 2: whp there are no two proper 3-colorings at distance greater than n / 100 d ¸ d 0, d 0 a sufficiently large constant

25. Proof Techniques – The Core Claim 3: Suppose that every vertex has the expected degree, and Claims 1 and 2 hold. Then the graph G is uniquely 3-colorable. Proof:  - the planted coloring. If not unique, 9 , dist( ,  )< n / 100 (Claim 1). U - set of disagreeing vertices.  v )  ( v ) ) v has d / 3 neighbors in U. |U| d | U |/ 6 – Contradicting Claim 2. V1V1 V3V3 V2V2

26. Proof Techniques – The Core This is whp the case when np > C k logn When np = O ( 1 ) – whp not the case Definition of Core H : v 2 H if v has at least np / 4 neighbors in G [ H ] in every other color class v has at most np / 10 neighbors outside of H. Claim 4: 9 Core H s.t. whp | H | ¸ (1-exp{-  ( np )}) n H is uniquely 3-colorable

27. Proof Techniques – The Core Corollary: (1-exp{-  ( np )}) n vertices are frozen in every proper 3-coloring Only one cluster of exponential size V1V1 V3V3 V2V2 V1V1 V2V2 V3V3

28. Moving to the Uniform Case A – a “bad” graph property (e.g. the graph has no big core)  – the expected number of proper k -colorings of random graph in the planted distribution Claim 5: Pr uniform [ A ] ·  ¢ Pr planted [ A ] Intuition: typically there are at most  ways to generate G in the planted model. Now use a union bound.

29. Moving to the Uniform Case A – “the graph has no big core” Claim 6: Pr planted [ A ] · e -exp{- C 1 } n Claim 7:  · e exp{- C 2 } n, C 2 > C 1 Corollary: Pr uniform [ A ] = o ( 1 ) There exists no proper 3-coloring w.r.t which there exists a big core

30. Algorithmic Perspective - Analysis Typically, uniform graphs have a big core Two more facts needed for the analysis: Claim 1 in the uniform case Logarithmic size components in G [ V \ H ] Both properties hold w.p. 1-1 / poly ( n ) in the planted model - cannot use “union bound” Solution: analyze directly the uniform distribution Difficulty: edges are strongly dependent Solution: careful, non-trivial, counting argument

31. Proof Techniques – a sampler Proof Technique - using the union bound: Fix a set U of t vertices, t · n /100 Fix a “bad” graph on the vertices in U Upper bound the probability of such graph Sum over all possible choices for U Claim 1: whp there is no subgraph H of G s.t. | V ( H )| C | H |/ 10 G n, m | { k –colorability }, m=Cn, C some large constant.

32. Proof Techniques – a sampler There are ways to choose the set U There are ways to fix the “bad” graph Pr[ the bad graph is a subgraph of G ] = If G was in G n, p then = p dt / 10 Recall G n, m | { k –colorability } In the uniform distribution, what is p ?

33. Proof Techniques – a sampler Let p be s.t. We shall Prove: Pr uniform [ fixed subgraph with r edges ] · ( 6p ) r Assume we did, then the probability of the claim not holding is at most The expected number of edges had the graph been sampled according to the planted distribution p=3m/n2p=3m/n2

34. Proof Techniques – a sampler Our goal: prove that Pr uniform [ fixed subgraph with r edges ] · ( 6p ) r Fix “bad” subgrpah B V 1 = G i contains B as subgraph V 2 =all k -colorable graphs with m edges G1G2...G1G2... H1H2...H1H2...

35. Proof Techniques – a sampler  1  – average degree of left side  2 – average degree of right side Double counting: | V 1 |  1 = | V 2 |  2 | V 1 | · | V 2 |(  2    1 ) We show a bipartite graph s.t. (  2    1 )=( 6p ) r ( G i, H j ) 2 E if G i can be obtained from H j using the following procedure: G1G2...G1G2... H1H2...H1H2...

36. Proof Techniques – a sampler Remove edges of B from G i, and replace them arbitrarily s.t. at least one k -coloring is respected Denote Observe that: Therefore,

37. Further Research Loose Rigorously analyze Survey Propagation on near- threshold formulas/graphs First step – analyze Survey Propagation on Planted instances Prove the near-threshold clustering phenomena Rigorously analyze message passing algorithms Analyze instances with an arbitrary constant (above the threshold) density