Improved bounds for sampling colorings

Abstract

We consider the problem of sampling uniformly at random from the set of proper k-colorings of a graph with maximum degree Δ. Our main result is the design of a simple Markov chain that converges in O(nk log n) time to the desired distribution when k>116Δ.

Path coupling without contraction

“…The coupling C for M is trivially extended to a coupling C r for M r and it easily follows that β(M r , C r ) β(M, C) r and σ 2 (M r , C r ) = σ 2 r (M, C). The standard analysis now yields mixing time bounds for M r and therefore M. Vigoda suggests a similar approach in [15] to claim a mixing time for his chain on graph colourings in the β = 1 case. We discuss this in Section 5.2 below.…”

“…Where this theorem does not apply, we give a modification of this idea which will often be applicable. We illustrate our methods with examples taken from the literature [2,4,10,15] which have made use of the non-expansion case of path coupling. In all these examples we improve the bound on mixing time established in the source paper.…”

Algorithms to approximately count and sample conforming colorings of graphs

“…The WSK chain chooses a vertex v and a color c at random and tries to recolor v with c. If it cannot be recolored, this is because there is a neighbor colored c, so we can instead consider the bipartite (c, c(v)) component, where c(v) is the current color of v. The WSK chain allows moves that swap the two colors on this whole component. Vigoda studied this chain and showed that a weighted version of the chain is rapidly mixing when k ≥ 11∆/6, where ∆ is the maximum degree [24]. In general variants of the WSK chain have proven difficult to analyze, although we will discuss a variant of this algorithm.…”

Torpid mixing of the Wang–Swendsen–Kotecký algorithm for sampling colorings

“…Two particular Markov chains are typically considered: the Wang-Swendsen-Kotecký (WSK) algorithm [12,13] and the simple Glauber dynamics [5]. Recently the second author proved that both of these chains are rapidly mixing when k > 11∆/6 [11]. We study how far these positive results can be extended in general and give examples of when the algorithms are torpidly mixing.…”