Mixing of Markov Chains for Independent Sets on Chordal Graphs with Bounded Separators

Abstract

We prove rapid mixing of well-known Markov chains for the hardcore model on a new graph class, the class of chordal graphs with a bound on minimal separator size. In the hardcore model, for a given graph G and a fugacity parameter \(\lambda \in \mathbb {R}^+\), the goal is to produce an independent set S of G with probability proportional to \(\lambda ^{|S|}\). In general graphs and arbitrary \(\lambda \), producing a sample from this distribution in polynomial time is provably difficult. However, natural Markov chains converge to the correct distribution for any graph, leading to the study of their mixing times for different graph classes. Rapid mixing for graphs of bounded degrees and a range of \(\lambda \)s dependent on the maximum degree has attracted attention since the 1990s. Recent results showed rapid mixing for arbitrary \(\lambda \) and two other classes of graphs: graphs of bounded treewidth and graphs of bounded bipartite pathwidth. In this work, we extend these results by showing rapid mixing in a new graph class, class of chordal graphs with bounded minimal separators. Graphs in this class have no bound on the vertex degrees, the treewidth, or the bipartite pathwidth. Similar to the results dealing with bounded treewidth and with bounded bipartite pathwidth, we prove rapid mixing using the canonical paths technique. However, unlike in the previous works, we need to process the data using a non-linear, tree-like, approach.