Strong spatial mixing of list coloring of graphs

Abstract

The property of spatial mixing and strong spatial mixing in spin systems has been of interest because of its implications on uniqueness of Gibbs measures on infinite graphs and efficient approximation of counting problems that are otherwise known to be #P hard. In the context of coloring, strong spatial mixing has been established for regular trees in [GS11] when q ≥ α * ∆ + 1 where q the number of colors, ∆ is the degree and α * = 1.763.. is the unique solution to xe −1/x = 1. It has also been established in [GMP05] for bounded degree lattice graphs whenever q ≥ α * ∆ − β for some constant β, where ∆ is the maximum vertex degree of the graph. The latter uses a technique based on recursively constructed coupling of Markov chains whereas the former is based on establishing decay of correlations on the tree. We establish strong spatial mixing of list colorings on arbitrary bounded degree triangle-free graphs whenever the size of the list of each vertex v is at least α∆(v)+ β where ∆(v) is the degree of vertex v and α > α * and β is a constant that only depends on α. We do this by proving the decay of correlations via recursive contraction of the distance between the marginals measured with respect to a suitably chosen error function.

A Deterministic Algorithm for Counting Colorings with 2-Delta Colors

“…Note that this condition refers only to real non-negativew, and hence is amenable to analysis via standard combinatorial tools. Indeed, our proofs that the conditions on q and ∆ in Theorems 1.1 and 1.2 imply this niceness condition are very similar to probabilistic arguments used by Gamarnik et al [19] to establish the property of "strong spatial mixing" (in the special casew = 0). We emphasize that this is the only place in our analysis where the lower bounds on q are used.…”

“…We restate Lemma 3.4 here for convenience. Next we show that the upper bound of 1 d+2 , where d is the number of unpinned neighbors of u in G, holds conditioned on every coloring of the neighbors of the (unpinned) neighbors of u, by following a similar path as in [19]. Consider any valid coloring 8 σ of the vertices at distance two from u.…”

“…unique solution to xe −1/x = 1 and β is a constant depending on α, and under the assumption that G is triangle-free (see also [20,21] for similar results on restricted classes of graphs). However, as discussed in [19], this strong spatial mixing result unfortunately does not lead to a deterministic algorithm. 2 The newer technique of polynomial interpolation, pioneered by Barvinok [3], has also recently been brought to bear on counting colorings.…”

“…For the special case ∆ = 3, Lu et al [32] give a correlation decay algorithm for counting 4-colorings. Furthermore, Gamarnik, Katz and Misra [19] establish the related property of "strong spatial mixing" under the weaker condition q ≥ α∆ + β for any constant α > α , where α ≈ 1.7633 is the 1 In this case, the notion of an FPRAS is replaced by that of a fully polynomial time approximation scheme, or FPTAS. An FPTAS for q-colorings of graphs of maximum degree at most ∆ is an algorithm that given the graph G and an error parameter δ on the input, produces a (1 ± δ)-factor multiplicative approximation to the number of q-colorings of G in time poly(|G|, 1/δ) (the degree of the polynomial is allowed to depend upon the constants q and ∆).…”

“…In fact, as discussed later in section 1.3, our technique is capable of directly harnessing strong spatial mixing arguments used in the analysis of Markov chains for certain classes of graphs. As an example, we can exploit such an argument of Gamarnik, Katz and Misra [19] to improve the bound on q in Theorem 1.1 when the graph is triangle-free, for all but small values of ∆. (Recall that α ≈ 1.7633 is the unique positive solution of the equation xe −1/x = 1.)…”