Abstract
Motivated by the 'subgraphs world' view of the ferromagnetic Ising model, we analyse the mixing times of Glauber dynamics based on subset expansion expressions for classes of graph, hypergraph and matroid polynomials. With a canonical paths argument, we demonstrate that the chains defined within this framework mix rapidly upon graphs, hypergraphs and matroids of bounded tree-width.This extends known results on rapid mixing for the Tutte polynomial, adjacency-rank ($R_2$-)polynomial and interlace polynomial. In particular Glauber dynamics for the $R_2$-polynomial was known to mix rapidly on trees, which led to hope of rapid mixing on a wider class of graphs. We show that Glauber dynamics for a very wide class of polynomials mixes rapidly on graphs of bounded tree-width, including many cases in which the Glauber dynamics does not mix rapidly for all graphs. This demonstrates that rapid mixing on trees or bounded tree-width graphs does not offer strong evidence towards rapid mixing on all graphs.
Highlights
We analyse a subset-sampling Markov chain on graphs that is derived from subset expansion1 graph functions, which include many wellknown graph polynomials
We show that Glauber dynamics for a very wide class of polynomials mixes rapidly on graphs of bounded tree-width, including many cases in which the Glauber dynamics does not mix rapidly for all graphs
We show that when the weight function w of some subset expansion formula is strictly positive and λ-multiplicative, Glauber dynamics is rapidly mixing on graphs of bounded treewidth
Summary
We analyse a subset-sampling Markov chain on graphs (and later hypergraphs and matroids) that is derived from subset expansion graph functions, which include many wellknown graph polynomials. Our general approach is an extension of work by Ge and Stefankovic [26] (see an earlier version [27]), which showed that the Markov chain for the (soft-core) random cluster model — i.e. weighted according to (2) — mixes rapidly upon graphs of bounded tree-width. Our work started with the R2-polynomial; Ge and Stefankovic introduced both this polynomial and its associated Glauber dynamics in an attempt to devise an approximation scheme for #BIS, the problem of counting the number of independent sets in a bipartite graph Their adjacency-rank polynomial is defined for any G = (V, E) and parameters q, μ as. Using a combinatorial interpretation of rk applicable only to bipartite graphs, they showed that the edge subset Glauber dynamics (using the weighting in (3)) mixes rapidly on trees.
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