A Counterexample to rapid mixing of the Ge-Stefankovic Process

TL;DR

This paper demonstrates that the Markov chain proposed by Ge and Stefankovic for sampling independent sets in bipartite graphs does not mix rapidly, as its mixing time is exponential for certain graph families, challenging its efficiency.

Contribution

The paper provides the first counterexamples showing that the Ge-Stefankovic Markov chain does not always mix rapidly, contradicting previous assumptions about its efficiency.

Findings

Counterexamples with exponential mixing time

Challenges the effectiveness of the proposed sampling method

Implications for approximation algorithms for counting problems

Abstract

Ge and Stefankovic have recently introduced a novel two-variable graph polynomial. When specialised to a bipartite graphs G and evaluated at the point (1/2,1) this polynomial gives the number of independent sets in the graph. Inspired by this polynomial, they also introduced a Markov chain which, if rapidly mixing, would provide an efficient sampling procedure for independent sets in G. This sampling procedure in turn would imply the existence of efficient approximation algorithms for a number of significant counting problems whose complexity is so far unresolved. The proposed Markov chain is promising, in the sense that it overcomes the most obvious barrier to mixing. However, we show here, by exhibiting a sequence of counterexamples, that the mixing time of their Markov chain is exponential in the size of the input when the input is chosen from a particular infinite family of bipartite graphs.