Spatial mixing and the connective constant: Optimal bounds

We study the problem of deterministic approximate counting of matchings and independent sets in graphs of bounded connective constant. More generally, we consider the problem of evaluating the partition functions of the monomer-dimer model (which is defined as a weighted sum over all matchings where each matching is given a weight γ|V |−2|M| in terms of a fixed parameter γ called the monomer activity) and the hard core model (which is defined as a weighted sum over all independent sets where an independent set I is given a weight λ|I| in terms of a fixed parameter λ called the vertex activity). The connective constant is a natural measure of the average degree of a graph which has been studied extensively in combinatorics and mathematical physics, and can be bounded by a constant even for certain unbounded degree graphs such as those sampled from the sparse Erdős–Rényi model G(n, d/n). Our main technical contribution is to prove the best possible rates of decay of correlations in the natural probability distributions induced by both the hard core model and the monomer-dimer model in graphs with a given bound on the connective constant. These results on decay of correlations are obtained using a new framework based on the so-called message approach that has been extensively used recently to prove such results for bounded degree graphs. We then use these optimal decay of correlations results to obtain FPTASs for the two problems on graphs of bounded connective constant. In particular, for the monomer-dimer model, we give a deterministic FPTAS for the partition function on all graphs of bounded connective constant for any given value of the monomer activity. The best previously known deterministic algorithm was due to Bayati, Gamarnik, Katz, Nair and Tetali [STOC 2007], and gave the same runtime guarantees as our results but only for the case of bounded degree graphs. For the hard core model, we give an FPTAS for graphs of connective constant Δ whenever ∗Alistair Sinclair, Computer Science Division, UC Berkeley. Email: [email protected]. Supported in part by NSF grant CCF-1016896 and by the Simons Institute for the Theory of Computing. †Piyush Srivastava, Center for the Mathematics of Information, Caltech. Email: [email protected]. Supported by NSF grant CCF-1319745. This work was done while this author was a graduate student at UC Berkeley and was supported by NSF grant CCF-1016896. ‡Daniel Štefankovič, Department of Computer Science, University of Rochester. Email: [email protected]. Supported in part by NSF grant CCF-1016896. Part of this work was done while this author was visiting the Simons Institute for the Theory of Computing. §Yitong Yin, State Key Laboratory for Novel Software Technology, Nanjing University, China. Email: [email protected]. Supported by NSFC grants 61272081 and 61321491. Part of this work was done while this author was visiting UC Berkeley. the vertex activity λ < λc(Δ), where λc(Δ) ··= Δ (Δ−1)Δ+1 ; this result is optimal in the sense that an FPTAS for any λ > λc(Δ) would imply that NP=RP [Sly, FOCS 2010]. The previous best known result in this direction was a recent paper by a subset of the current authors [FOCS 2013], where the result was established under the suboptimal condition λ < λc(Δ + 1). Our techniques also allow us to improve upon known bounds for decay of correlations for the hard core model on various regular lattices, including those obtained by Restrepo, Shin, Vigoda and Tetali [FOCS 11] for the special case of Z using sophisticated numerically intensive methods tailored to that special case.