Counting Graph Colourings by using Sequences of Subgraphs

Counting Graph Colourings by using Sequences of Subgraphs Charilaos Efthymiou DIMAP University of Warwick DIMAP Summer School – July 2010

Counting l Problem: Problem Given Find the cardinality of the set of feasible solutions l Examples: Examples matching, independent sets, proper colourings, bin packing, SAT

Graph Colouring G=(V, E) Set of colours {1, …, k} (Proper) Colouring σ : V →{1, …, k} 1 and σ(v)≠σ(u) for every {v, u} E 1 1 2 1 3 Colours : {1, 2, 3, 4} 6 3 2 5 3 4

Counting Vs Sampling G=(V, E) Set of colours {1, …, k} Z(G, k): k) k-colouring of G Process: Choose u. a. r. a k-colouring of the graph Ei, j: “Vertices i, j receive different colour” pa: : Pr[ E 1, 6 in Ga] pb: Pr[ E 1, 4 in Gb] pc: Pr[ E 2, 6 in Gc] pd: Pr[ E 2, 5 in Gd] pe: Pr[ {6, 3} in Ge] Z(G, k)=|V|k pa pb pc pd pe 2 1 c d a 3 e 6 5 b 4

Counting l Exact counting is hard, l Valiant ‘ 79 P class l Approximate counting using Rapidly Mixing Markov Chains l Celebrated achievements: FPRAS for l l l Permanents: Jerrum, Sinclair 1989, Jerrum Sinclair Vigoda 2001 Volume of convex body: Dyer, Frieze Kannan 1991 Counting independent sets in degree-4 graphs: Luby and Vigoda 1997

Graph k-colourings l Maximum degree Δ § § § [Vigoda 99] k>11/6Δ – arbitrary graph [Mοssel & Sly 08] Random graphs with fixed expected degree d , with k>f(d) [Hayes, Vera & Vigoda] Planar Graphs k > Ω(Δ/log Δ)

For this talk… l l We propose algorithms which are not based on Markov Chains. Compute the corresponding probabilities directly. l Weakness: l l We only compute ε-approximation to log Z(G, k) Strength: l l Deterministic Explicit results for Gnp

Works for Det. Counting l Colourings l l Regular graphs with high girth Δ+1 -colours- PTAS l Bandyopadhyay, Gamarnik 2005 G with girth 4, 2. 8Δ-colours FPTAS l Gamarnik, Katz 2007 Sparse Random Graphs with number of colours that depend on the expected degree -PTAS l Efthymiou, Spirakis 2008. Graph matchings - FPTAS l l Bayatui, Gamarnik, Katz, Nair, Tetali 2007. Independent sets FPTAS l Weitz 2006

Easy Examples… l Trees l Compute each probability recursively! l DP - - For constant k the time-complexity is O(n) l Graphs of bounded treewidth l Graphs with number of k-colourings (proper & non-proper) that is O(nc)

Efficient computation of marginals – Correlation decay u v A B t C

3 -step algorithm l Compute Pr[Euv] on the “small” graph. l Prove independence from boundary conditions. l l Dobrushin’s Condition for Uniqueness of Gibbs measure Project to the initial graph.

Reduce computational load differently… u l 1 r 1 l 2 r 2 l 3 r 3 l 4 r 4 v A t B C

Reduce computational load differently… u l 1 r 1 l 2 r 2 l 3 r 3 l 4 r 4 v A B C

Implications on spatial mixing conditions u l 1 r 1 l 2 r 2 l 3 r 3 l 4 r 4 v A t L(l 3, t): Vertices outside red cycle B C t L(r 3, t): Vertices outside green cycle

Comparison with the first approach u l 1 r 1 l 2 r 2 l 3 r 3 l 4 r 4 v t L(l 3, t): Vertices outside red cycle

Theorem - Accuracy u l 1 r 1 l 2 r 2 l 3 r 3 l 4 r 4 v

Spatial Correlation decay G=(V, E) u BP(v, u) Product measure: Pq Pr[Disagreeing]=q Pr[Non-Disagreeing]=1 -q v “Path of disagreement between u & v”

Bounding Spatial Correlation decay :

Applications I – Sparse Gnp l The underlying graph is Gnp with expected degree d, d is fixed l l l Vertex set V={1, …, n} Each possible edge appear with probability p, independently of the others. Expected degree is d is fixed real, i. e. p=d/n The maximum degree is Θ(log n/ loglog n) Chromatic number Constant

Applications I – Sparse Gnp l Isolate Θ(log n) neighborhoods around u, v l l l Tree with additional Θ(log n) edges Computations by Dynamic Programming Using k≥(2+ε)d with probability 1 -n-Ω(1) we get a polynomial time, n-Ω(1)-approximation of log Z(Gnp, k).

Applications II – Locally α-dense graphs l G(V, E) is locally α-dense of bounded maximum degree Δ if l l For all {w 1, w 2} E w 2 has at most (1 -α)Δ neighbors which are not adjacent to w 1 α [0, 1] is a parameter of the model

Applications II – Locally dense graphs l For k>(2 -α)Δ we get a (log n)-Ω(1)-approximation of log Z(G, k), in polynomial time. l If, additionally, every Θ(log n) neighborhood of G has constant treewidth, then we get a n-Ω(1)approximation of log Z(G, k), in polynomial time.

Thank You!