A Counterexample to rapid mixing of the Ge-Stefankovic Process

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.

💡 Research Summary

The paper investigates the feasibility of efficiently approximating the number of independent sets in bipartite graphs, a problem known as #BIS. While exact counting is #P‑complete, the existence of a fully polynomial‑time randomized approximation scheme (FPRAS) for #BIS remains an open question. Traditional Markov‑chain Monte‑Carlo (MCMC) approaches that operate directly on vertex subsets fail on simple instances such as the complete bipartite graph because any walk from a left‑oriented independent set to a right‑oriented one must pass through the empty set, creating a bottleneck with exponentially small conductance.

Ge and Stefankovic recently introduced a two‑variable graph polynomial (R’2(G;\lambda,\mu)) whose evaluation at ((\lambda,\mu)=(1/2,1)) yields the exact number of independent sets in a bipartite graph (G). Inspired by this polynomial they defined a new Markov chain, the Ge‑Stefankovic (GS) Process, whose state space consists of edge subsets (\Omega = 2^{E}) rather than vertex subsets. The chain performs a “single‑bond flip”: at each step it toggles the presence of a single edge, making it a 1‑cautious chain (i.e., the Hamming distance between successive states is at most one). The stationary distribution (\pi{\Omega}) of this chain is precisely the marginal distribution induced by the consistency relation (\chi(I,A)) between vertex subsets (I\subseteq U) and edge subsets (A\subseteq E). The marginal on vertex subsets, (\pi_{\Sigma}), coincides with the distribution obtained by projecting a uniformly random independent set onto the left side (U).

The authors construct a family of bipartite graphs ({G_n}) that serve as counterexamples to rapid mixing. For each (n) they choose an integer (m) satisfying ((3/2)m \le 2^{n-1} < (3/2)m+1) and define three vertex groups: (U’ (|U’|=n)), (V (|V|=m)), and (U’’ (|U’’|=m)). Edges consist of a complete bipartite connection between (U’) and (V) together with a perfect matching (M) between (V) and (U’’). This structure yields a natural partition of the vertex‑subset space (\Sigma = 2^{U}) into (\Sigma_0 = {I: I\cap U’ = \emptyset}) and (\Sigma_1 = \Sigma\setminus\Sigma_0). Because there are (3^{m}) independent sets avoiding (U’) and ((2^{n}-1)2^{m}) that include at least one vertex of (U’), the stationary mass of (\Sigma_0) is a constant (\alpha) satisfying (2/5 < \alpha \le 1/2); thus (\Sigma_0) and (\Sigma_1) are nearly balanced.

For the edge‑subset space (\Omega) they define a weight function (w(A)=|A\cap M|) and partition (\Omega) into (\Omega_0 = {A: w(A)\le 5m/12}) and (\Omega_1 = \Omega\setminus\Omega_0). The crucial observation is that any transition from (\Sigma_0) to (\Sigma_1) (or from (\Omega_0) to (\Omega_1)) must pass through a “barrier” set \