A Counterexample to rapid mixing of the Ge-Stefankovic Process

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.