Approximation algorithms for two-state anti-ferromagnetic spin systems on bounded degree graphs

In a seminal paper [10], Weitz gave a deterministic fully polynomial approximation scheme for counting exponentially weighted independent sets (which is the same as approximating the partition function of the hard-core model from statistical physics) in graphs of degree at most d, up to the critical activity for the uniqueness of the Gibbs measure on the in nite d-regular tree. More recently Sly [8] (see also [1]) showed that this is optimal in the sense that if there is an FPRAS for the hard-core partition function on graphs of maximum degree d for activities larger than the critical activity on the in nite d-regular tree then NP = RP. In this paper we extend Weitz's approach to derive a deterministic fully polynomial approximation scheme for the partition function of general two-state anti-ferromagnetic spin systems on graphs of maximum degree d, up to the corresponding critical point on the d-regular tree. The main ingredient of our result is a proof that for two-state anti-ferromagnetic spin systems on the d-regular tree, weak spatial mixing implies strong spatial mixing. This in turn uses a message-decay argument which extends a similar approach proposed recently for the hard-core model by Restrepo et al [7] to the case of general two-state anti-ferromagnetic spin systems

A Simple FPTAS for Counting Edge Covers

An edge cover of a graph is a set of edges such that every vertex has at least an adjacent edge in it. Previously, approximation algorithm for counting edge covers is only known for 3 regular graphs and it is randomized. We design a very simple deterministic fully polynomial-time approximation scheme (FPTAS) for counting the number of edge covers for any graph. Our main technique is correlation decay, which is a powerful tool to design FPTAS for counting problems. In order to get FPTAS for general graphs without degree bound, we make use of a stronger notion called computationally efficient correlation decay, which is introduced in [Li, Lu, Yin SODA 2012].Comment: To appear in SODA 201

Structure learning of antiferromagnetic Ising models

In this paper we investigate the computational complexity of learning the graph structure underlying a discrete undirected graphical model from i.i.d. samples. We first observe that the notoriously difficult problem of learning parities with noise can be captured as a special case of learning graphical models. This leads to an unconditional computational lower bound of $\Omega (p^{d/2})$ for learning general graphical models on $p$ nodes of maximum degree $d$, for the class of so-called statistical algorithms recently introduced by Feldman et al (2013). The lower bound suggests that the $O(p^d)$ runtime required to exhaustively search over neighborhoods cannot be significantly improved without restricting the class of models. Aside from structural assumptions on the graph such as it being a tree, hypertree, tree-like, etc., many recent papers on structure learning assume that the model has the correlation decay property. Indeed, focusing on ferromagnetic Ising models, Bento and Montanari (2009) showed that all known low-complexity algorithms fail to learn simple graphs when the interaction strength exceeds a number related to the correlation decay threshold. Our second set of results gives a class of repelling (antiferromagnetic) models that have the opposite behavior: very strong interaction allows efficient learning in time $O(p^2)$. We provide an algorithm whose performance interpolates between $O(p^2)$ and $O(p^{d+2})$ depending on the strength of the repulsion.Comment: 15 pages. NIPS 201

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

Sublinear-Time Algorithms for Monomer-Dimer Systems on Bounded Degree Graphs

For a graph $G$, let $Z(G,\lambda)$ be the partition function of the monomer-dimer system defined by $\sum_k m_k(G)\lambda^k$, where $m_k(G)$ is the number of matchings of size $k$ in $G$. We consider graphs of bounded degree and develop a sublinear-time algorithm for estimating $\log Z(G,\lambda)$ at an arbitrary value $\lambda>0$ within additive error $\epsilon n$ with high probability. The query complexity of our algorithm does not depend on the size of $G$ and is polynomial in $1/\epsilon$, and we also provide a lower bound quadratic in $1/\epsilon$ for this problem. This is the first analysis of a sublinear-time approximation algorithm for a # P-complete problem. Our approach is based on the correlation decay of the Gibbs distribution associated with $Z(G,\lambda)$. We show that our algorithm approximates the probability for a vertex to be covered by a matching, sampled according to this Gibbs distribution, in a near-optimal sublinear time. We extend our results to approximate the average size and the entropy of such a matching within an additive error with high probability, where again the query complexity is polynomial in $1/\epsilon$ and the lower bound is quadratic in $1/\epsilon$. Our algorithms are simple to implement and of practical use when dealing with massive datasets. Our results extend to other systems where the correlation decay is known to hold as for the independent set problem up to the critical activity

FPTAS for Hardcore and Ising Models on Hypergraphs

Hardcore and Ising models are two most important families of two state spin systems in statistic physics. Partition function of spin systems is the center concept in statistic physics which connects microscopic particles and their interactions with their macroscopic and statistical properties of materials such as energy, entropy, ferromagnetism, etc. If each local interaction of the system involves only two particles, the system can be described by a graph. In this case, fully polynomial-time approximation scheme (FPTAS) for computing the partition function of both hardcore and anti-ferromagnetic Ising model was designed up to the uniqueness condition of the system. These result are the best possible since approximately computing the partition function beyond this threshold is NP-hard. In this paper, we generalize these results to general physics systems, where each local interaction may involves multiple particles. Such systems are described by hypergraphs. For hardcore model, we also provide FPTAS up to the uniqueness condition, and for anti-ferromagnetic Ising model, we obtain FPTAS where a slightly stronger condition holds

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