Rapid mixing of the hardcore Glauber dynamics and other Markov chains in bounded-treewidth graphs
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We give a new rapid mixing result for a natural random walk on the independent sets of an input graph G. Rapid mixing is of interest in approximately sampling a structure, over some underlying set or graph, from some target distribution. In the case of independent sets, we show that when G has bounded treewidth, this random walk – known as the hardcore Glauber dynamics – mixes rapidly for all values of the standard parameter λ > 0, giving a simple alternative to existing sampling algorithms for these structures. We also show rapid mixing for Markov chains on dominating sets and b-edge covers (for fixed b≥ 1 and λ > 0) in the case where treewidth is bounded, and for Markov chains on the b-matchings (for fixed b ≥ 1 and λ > 0), the maximal independent sets, and the maximal b-matchings of a graph (for fixed b ≥ 1), in the case where carving width is bounded. We prove our results by developing a divide-and-conquer framework using the well-known multicommodity flows technique. Using this technique, we additionally show that a similar dynamics on the k-angulations of a convex set of n points mixes in quasipolynomial time for all k ≥ 3. (McShine and Tetali gave a stronger result in the special case k = 3.) Our technique also allows us to strengthen existing results by Dyer, Goldberg, and Jerrum and by Heinrich for the Glauber dynamics on the q-colorings of G on graphs of bounded carving width, when q ≥Δ + 2 is bounded. Specifically, our technique yields an improvement in the dependence on treewidth when Δ < 2t or when q < 4t and Δ < t^2. We additionally show that the Glauber dynamics on the partial q-colorings of G mix rapidly for all λ > 0 when q ≥Δ + 2 is bounded.
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