Spatial mixing and approximation algorithms for graphs with bounded connective constant

The hard core model in statistical physics is a probability distribution on independent sets in a graph in which the weight of any independent set I is proportional to lambda^(|I|), where lambda > 0 is the vertex activity. We show that there is an intimate connection between the connective constant of a graph and the phenomenon of strong spatial mixing (decay of correlations) for the hard core model; specifically, we prove that the hard core model with vertex activity lambda < lambda_c(Delta + 1) exhibits strong spatial mixing on any graph of connective constant Delta, irrespective of its maximum degree, and hence derive an FPTAS for the partition function of the hard core model on such graphs. Here lambda_c(d) := d^d/(d-1)^(d+1) is the critical activity for the uniqueness of the Gibbs measure of the hard core model on the infinite d-ary tree. As an application, we show that the partition function can be efficiently approximated with high probability on graphs drawn from the random graph model G(n,d/n) for all lambda < e/d, even though the maximum degree of such graphs is unbounded with high probability. We also improve upon Weitz's bounds for strong spatial mixing on bounded degree graphs (Weitz, 2006) by providing a computationally simple method which uses known estimates of the connective constant of a lattice to obtain bounds on the vertex activities lambda for which the hard core model on the lattice exhibits strong spatial mixing. Using this framework, we improve upon these bounds for several lattices including the Cartesian lattice in dimensions 3 and higher. Our techniques also allow us to relate the threshold for the uniqueness of the Gibbs measure on a general tree to its branching factor (Lyons, 1989).Comment: 26 pages. In October 2014, this paper was superseded by arxiv:1410.2595. Before that, an extended abstract of this paper appeared in Proc. IEEE Symposium on the Foundations of Computer Science (FOCS), 2013, pp. 300-30

Approximate Capacities of Two-Dimensional Codes by Spatial Mixing

We apply several state-of-the-art techniques developed in recent advances of counting algorithms and statistical physics to study the spatial mixing property of the two-dimensional codes arising from local hard (independent set) constraints, including: hard-square, hard-hexagon, read/write isolated memory (RWIM), and non-attacking kings (NAK). For these constraints, the strong spatial mixing would imply the existence of polynomial-time approximation scheme (PTAS) for computing the capacity. It was previously known for the hard-square constraint the existence of strong spatial mixing and PTAS. We show the existence of strong spatial mixing for hard-hexagon and RWIM constraints by establishing the strong spatial mixing along self-avoiding walks, and consequently we give PTAS for computing the capacities of these codes. We also show that for the NAK constraint, the strong spatial mixing does not hold along self-avoiding walks

Spatial Mixing of Coloring Random Graphs

We study the strong spatial mixing (decay of correlation) property of proper $q$-colorings of random graph $G(n, d/n)$ with a fixed $d$. The strong spatial mixing of coloring and related models have been extensively studied on graphs with bounded maximum degree. However, for typical classes of graphs with bounded average degree, such as $G(n, d/n)$, an easy counterexample shows that colorings do not exhibit strong spatial mixing with high probability. Nevertheless, we show that for $q\ge\alpha d+\beta$ with $\alpha>2$ and sufficiently large $\beta=O(1)$, with high probability proper $q$-colorings of random graph $G(n, d/n)$ exhibit strong spatial mixing with respect to an arbitrarily fixed vertex. This is the first strong spatial mixing result for colorings of graphs with unbounded maximum degree. Our analysis of strong spatial mixing establishes a block-wise correlation decay instead of the standard point-wise decay, which may be of interest by itself, especially for graphs with unbounded degree

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 fully polynomial time approximation schemes (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 et al. (Proc. 39th ACM Symp. Theory Comput., pp. 122–127, 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 λ_c(Δ) would imply that NP=RP (Sly and Sun, Ann. Probab. 42(6):2383–2416, 2014). The previous best known result in this direction was in a recent manuscript by a subset of the current authors (Proc. 54th IEEE Symp. Found. Comput. Sci., pp 300–309, 2013), where the result was established under the sub-optimal condition λ < λ_c(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 et al. (Probab Theory Relat Fields 156(1–2):75–99, 2013) for the special case of Z^2 using sophisticated numerically intensive methods tailored to that special case

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 γ^(|I|) in terms of a fixed parameter γ 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 Erdös-Rényi model (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 λ_c(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^2 using sophisticated numerically intensive methods tailored to that special case

Counting hypergraph matchings up to uniqueness threshold

We study the problem of approximately counting matchings in hypergraphs of bounded maximum degree and maximum size of hyperedges. With an activity parameter $\lambda$, each matching $M$ is assigned a weight $\lambda^{|M|}$. The counting problem is formulated as computing a partition function that gives the sum of the weights of all matchings in a hypergraph. This problem unifies two extensively studied statistical physics models in approximate counting: the hardcore model (graph independent sets) and the monomer-dimer model (graph matchings). For this model, the critical activity $\lambda_c= \frac{d^d}{k (d-1)^{d+1}}$ is the threshold for the uniqueness of Gibbs measures on the infinite $(d+1)$-uniform $(k+1)$-regular hypertree. Consider hypergraphs of maximum degree at most $k+1$ and maximum size of hyperedges at most $d+1$. We show that when $\lambda < \lambda_c$, there is an FPTAS for computing the partition function; and when $\lambda = \lambda_c$, there is a PTAS for computing the log-partition function. These algorithms are based on the decay of correlation (strong spatial mixing) property of Gibbs distributions. When $\lambda > 2\lambda_c$, there is no PRAS for the partition function or the log-partition function unless NP$=$RP. Towards obtaining a sharp transition of computational complexity of approximate counting, we study the local convergence from a sequence of finite hypergraphs to the infinite lattice with specified symmetry. We show a surprising connection between the local convergence and the reversibility of a natural random walk. This leads us to a barrier for the hardness result: The non-uniqueness of infinite Gibbs measure is not realizable by any finite gadgets

The complexity of approximating the matching polynomial in the complex plane

We study the problem of approximating the value of the matching polynomial on graphs with edge parameter $\gamma$, where $\gamma$ takes arbitrary values in the complex plane. When $\gamma$ is a positive real, Jerrum and Sinclair showed that the problem admits an FPRAS on general graphs. For general complex values of $\gamma$, Patel and Regts, building on methods developed by Barvinok, showed that the problem admits an FPTAS on graphs of maximum degree $\Delta$ as long as $\gamma$ is not a negative real number less than or equal to $-1/(4(\Delta-1))$. Our first main result completes the picture for the approximability of the matching polynomial on bounded degree graphs. We show that for all $\Delta\geq 3$ and all real $\gamma$ less than $-1/(4(\Delta-1))$, the problem of approximating the value of the matching polynomial on graphs of maximum degree $\Delta$ with edge parameter $\gamma$ is #P-hard. We then explore whether the maximum degree parameter can be replaced by the connective constant. Sinclair et al. showed that for positive real $\gamma$ it is possible to approximate the value of the matching polynomial using a correlation decay algorithm on graphs with bounded connective constant (and potentially unbounded maximum degree). We first show that this result does not extend in general in the complex plane; in particular, the problem is #P-hard on graphs with bounded connective constant for a dense set of $\gamma$ values on the negative real axis. Nevertheless, we show that the result does extend for any complex value $\gamma$ that does not lie on the negative real axis. Our analysis accounts for complex values of $\gamma$ using geodesic distances in the complex plane in the metric defined by an appropriate density function