Fast Sampling of b-Matchings and b-Edge Covers
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For an integer b ≥ 1, a b-matching (resp. b-edge cover) of a graph G=(V,E) is a subset S⊆ E of edges such that every vertex is incident with at most (resp. at least) b edges from S. We prove that for any b ≥ 1 the simple Glauber dynamics for sampling (weighted) b-matchings and b-edge covers mixes in O(nlog n) time on all n-vertex bounded-degree graphs. This significantly improves upon previous results which have worse running time and only work for b-matchings with b ≤ 7 and for b-edge covers with b ≤ 2. More generally, we prove spectral independence for a broad class of binary symmetric Holant problems with log-concave signatures, including b-matchings, b-edge covers, and antiferromagnetic 2-spin edge models. We hence deduce optimal mixing time of the Glauber dynamics from spectral independence. The core of our proof is a recursive coupling inspired by (Chen and Zhang '23) which upper bounds the Wasserstein W_1 distance between distributions under different pinnings. Using a similar method, we also obtain the optimal O(nlog n) mixing time of the Glauber dynamics for the hardcore model on n-vertex bounded-degree claw-free graphs, for any fugacity λ. This improves over previous works which have at least cubic dependence on n.
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