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 In October 2014, this paper was superseded by arxiv:1410.2595.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 sampling and counting for spin models in graphs

En aquest treball abordem els problemes de mostreig i comptatge aproximat en models d'espins en grafs, recopilant els resultats més significatius de l'àrea i introduïnt els coneixements previs necessaris del camp de la física estadística. En particular, prestem especial atenció als mètodes generals de disseny d'algorismes desenvolupats per Weitz i Barvinok, així com els avenços recents en matèria de comptatge i mostreig de conjunts independents de mida donada. Així mateix, discutim com es podrien adaptar aquests arguments als problemes de comptatge i mostreig de coloracions amb les mides de cada color fixades, explicant amb detall la línia de recerca actual que estem duent a terme.En este trabajo abordamos los problemas de muestreo y conteo aproximado en modelos de espines en grafos, recopilando los resultados más significativos del campo e introduciendo el conocimiento previo necesario del área de la física estadística. En particular, prestamos especial atención a los métodos generales de diseño de algorismos desarrollados por Weitz y Barvinok, así como a los avances recientes en cuanto al conteo y muestreo de conjuntos independientes de un tamaño dado. Así mismo, discutimos cómo se podrían adaptar estos argumentos al problema de contar y muestrear coloraciones con el tamaño de cada color fijo, explicando en detalle la línea de investigación que estamos llevando a cabo actualmente.We approach the problems of approximate sampling and counting in spin models on graphs, surveying the most significant results in the area and introducing the necessary background from statistical physics. We pay particular attention to the general algorithm design frameworks developed by Weitz and Barvinok, as well as to the newer results on counting and sampling independent sets of given size. In addition, we discuss the adaptation of the arguments behind these results to count and sample colorings with fixed color sizes, explaining in detail the current research line we are undertaking.Outgoin

Sampling in Potts Model on Sparse Random Graphs

We study the problem of sampling almost uniform proper q-colorings in sparse Erdos-Renyi random graphs G(n,d/n), a research initiated by Dyer, Flaxman, Frieze and Vigoda [Dyer et al., RANDOM STRUCT ALGOR, 2006]. We obtain a fully polynomial time almost uniform sampler (FPAUS) for the problem provided q>3d+4, improving the current best bound q>5.5d [Efthymiou, SODA, 2014]. Our sampling algorithm works for more generalized models and broader family of sparse graphs. It is an efficient sampler (in the same sense of FPAUS) for anti-ferromagnetic Potts model with activity 03(1-b)d+4. We further identify a family of sparse graphs to which all these results can be extended. This family of graphs is characterized by the notion of contraction function, which is a new measure of the average degree in graphs

Sampling in Uniqueness from the Potts and Random-Cluster Models on Random Regular Graphs

We consider the problem of sampling from the Potts model on random regular graphs. It is conjectured that sampling is possible when the temperature of the model is in the so-called uniqueness regime of the regular tree, but positive algorithmic results have been for the most part elusive. In this paper, for all integers q >= 3 and Delta >= 3, we develop algorithms that produce samples within error o(1) from the q-state Potts model on random Delta-regular graphs, whenever the temperature is in uniqueness, for both the ferromagnetic and antiferromagnetic cases. The algorithm for the antiferromagnetic Potts model is based on iteratively adding the edges of the graph and resampling a bichromatic class that contains the endpoints of the newly added edge. Key to the algorithm is how to perform the resampling step efficiently since bichromatic classes can potentially induce linear-sized components. To this end, we exploit the tree uniqueness to show that the average growth of bichromatic components is typically small, which allows us to use correlation decay algorithms for the resampling step. While the precise uniqueness threshold on the tree is not known for general values of q and Delta in the antiferromagnetic case, our algorithm works throughout uniqueness regardless of its value. In the case of the ferromagnetic Potts model, we are able to simplify the algorithm significantly by utilising the random-cluster representation of the model. In particular, we demonstrate that a percolation-type algorithm succeeds in sampling from the random-cluster model with parameters p,q on random Delta-regular graphs for all values of q >= 1 and p<p_c(q,Delta), where p_c(q,Delta) corresponds to a uniqueness threshold for the model on the Delta-regular tree. When restricted to integer values of q, this yields a simplified algorithm for the ferromagnetic Potts model on random Delta-regular graphs

FPTAS for #BIS with Degree Bounds on One Side

Counting the number of independent sets for a bipartite graph (#BIS) plays a crucial role in the study of approximate counting. It has been conjectured that there is no fully polynomial-time (randomized) approximation scheme (FPTAS/FPRAS) for #BIS, and it was proved that the problem for instances with a maximum degree of $6$ is already as hard as the general problem. In this paper, we obtain a surprising tractability result for a family of #BIS instances. We design a very simple deterministic fully polynomial-time approximation scheme (FPTAS) for #BIS when the maximum degree for one side is no larger than $5$. There is no restriction for the degrees on the other side, which do not even have to be bounded by a constant. Previously, FPTAS was only known for instances with a maximum degree of $5$ for both sides.Comment: 15 pages, to appear in STOC 2015; Improved presentations from previous version

Sampling grid colourings with fewer colours

We provide an optimally mixing Markov chain for 6-colourings of the square grid. Furthermore, this implies that the uniform distribution on the set of such colourings has strong spatial mixing. 4 and 5 are now the only remaining values of k for which it is not known whether there exists a rapidly mixing Markov chain for k-colourings of the square grid

Spatial Mixing and Non-local Markov chains

We consider spin systems with nearest-neighbor interactions on an $n$-vertex $d$-dimensional cube of the integer lattice graph $\mathbb{Z}^d$. We study the effects that exponential decay with distance of spin correlations, specifically the strong spatial mixing condition (SSM), has on the rate of convergence to equilibrium distribution of non-local Markov chains. We prove that SSM implies $O(\log n)$ mixing of a block dynamics whose steps can be implemented efficiently. We then develop a methodology, consisting of several new comparison inequalities concerning various block dynamics, that allow us to extend this result to other non-local dynamics. As a first application of our method we prove that, if SSM holds, then the relaxation time (i.e., the inverse spectral gap) of general block dynamics is $O(r)$, where $r$ is the number of blocks. A second application of our technology concerns the Swendsen-Wang dynamics for the ferromagnetic Ising and Potts models. We show that SSM implies an $O(1)$ bound for the relaxation time. As a by-product of this implication we observe that the relaxation time of the Swendsen-Wang dynamics in square boxes of $\mathbb{Z}^2$ is $O(1)$ throughout the subcritical regime of the $q$-state Potts model, for all $q \ge 2$. We also prove that for monotone spin systems SSM implies that the mixing time of systematic scan dynamics is $O(\log n (\log \log n)^2)$. Systematic scan dynamics are widely employed in practice but have proved hard to analyze. Our proofs use a variety of techniques for the analysis of Markov chains including coupling, functional analysis and linear algebra

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

Approximate counting via complex zero-free regions and spectral independence

This thesis investigates fundamental problems in approximate counting that arise in the field of statistical mechanics. Building upon recent advancements in the area, our research aims to enhance our understanding of the computational complexity of sampling from the Ising and Potts models, as well as the random $k$-SAT model. The $q$-state Potts model is a spin model in which each particle is randomly assigned a spin (out of $q$ possible spins), where the probability of a certain assignment depends on how many adjacent particles present the same spin. The edge interaction of the model is a parameter that quantifies the strength of interaction between two adjacent particles. The Ising model corresponds to the Potts model with $q = 2$. Sampling from these models is inherently connected to approximating the partition function of the model, a graph polynomial that encodes several aggregate thermodynamic properties of the system. In addition to classical connections with quantum computing and phase transitions in statistical physics, recent work in approximate counting has shown that the behaviour in the complex plane of these partition functions, and more precisely the location of zeros, is strongly connected with the complexity of the approximation problem, even for positive real-valued parameters. Thus, following this trend in both statistical physics and algorithmic research, we allow the edge interaction to be any complex number. First, we study the complexity of approximating the partition function of the $q$-state Potts model and the closely related Tutte polynomial for complex values of the underlying parameters. Previous work in the complex plane by Goldberg and Guo focused on $q=2$; for $q>2$, the behaviour in the complex plane is not as well understood and most work applies only to the real-valued Tutte plane. Our main result is a complete classification of the complexity of the approximation problems for all non-real values of the parameters, by establishing \#P-hardness results that apply even when restricted to planar graphs. Our techniques apply to all $q\geq 2$ and further complement/refine previous results both for the Ising model and the Tutte plane, answering in particular a question raised by Bordewich, Freedman, Lov\'{a}sz and Welsh in the context of quantum computations. Secondly, we investigate the complexity of approximating the partition function \ising(G; \beta) of the Ising model in terms of the relation between the edge interaction $\beta$ and a parameter $\Delta$ which is an upper bound on the maximum degree of the input graph $G$. In this thesis we establish both new tractability and inapproximability results. Our tractability results show that \ising(-; \beta) has an FPTAS when $\beta \in \mathbb{C}$ and $\lvert \beta - 1 \rvert / \lvert \beta + 1 \rvert 1 / \sqrt{\Delta - 1}$. These are the first results to show intractability of approximating \ising(-, \beta) on bounded degree graphs with complex $\beta$. Moreover, we demonstrate situations in which zeros of the partition function imply hardness of approximation in the Ising model. Finally, we exploit the recently successful framework of spectral independence to analyse the mixing time of a Markov chain, and we apply it in order to sample satisfying assignments of $k$-CNF formulas. Our analysis leads to a nearly linear-time algorithm to approximately sample satisfying assignments in the random $k$-SAT model when the density of the random formula $\alpha=m/n$ scales exponentially with $k$, where $n$ is the number of variables and $m$ is the number of clauses. The best previously known sampling algorithm for the random $k$-SAT model applies when the density $\alpha=m/n$ of the formula is less than $2^{k/300}$ and runs in time $n^{\exp(\Theta(k))}$. Our algorithm achieves a significantly faster running time of $n^{1 + o_k(1)}$ and samples satisfying assignments up to density $\alpha\leq 2^{0.039 k}$. The main challenge in our setting is the presence of many variables with unbounded degree, which causes significant correlations within the formula and impedes the application of relevant Markov chain methods from the bounded-degree setting