Paper Info
Title
Spatial mixing and the connective constant: optimal bounds
Abstract
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 Erd&odblac;s-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 the vertex activity λ < λc(Δ), where λc(Δ): = [EQUATION]; 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 sub-optimal 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 Z2 using sophisticated numerically intensive methods tailored to that special case.
Year
DOI
Venue
2014
10.1007/s00440-016-0708-2
SODA
Keywords
DocType
Volume
82B20, 60J10, 68W25, 68W40
Journal
abs/1410.2595
Issue
ISSN
ISBN
1-2
1432-2064
978-1-61197-433-1
Citations 
PageRank 
References 
9
0.60
26
Authors
4
Name
Order
Citations
PageRank
Alistair Sinclair11506308.40
Piyush Srivastava2654.55
Daniel Štefankovic3221.43
Yitong Yin421323.42