In this paper we show a new way of constructing deterministic polynomial-time approximation algorithms for computing complex-valued evaluations of a large class of graph polynomials on bounded degree graphs. In particular, our approach works for the Tutte polynomial and independence polynomial, as well as partition functions of complex-valued spin and edge-coloring models.More specifically, we define a large class of graph polynomials C and show that if p ∈ C and there is a disk D centered at zero in the complex plane such that p(G) does not vanish on D for all bounded degree graphs G, then for each z in the interior of D there exists a deterministic polynomial-time approximation algorithm for evaluating p(G) at z. This gives an explicit connection between absence of zeros of graph polynomials and the existence of efficient approximation algorithms, allowing us to show new relationships between well-known conjectures.Our work builds on a recent line of work initiated by Barvinok [2,3,4,5], which provides a new algorithmic approach besides the existing Markov chain Monte Carlo method and the correlation decay method for these types of problems.

A conjecture of Sokal [23] regarding the domain of non-vanishing for independence polynomials of graphs, states that given any natural number ∆ ≥ 3, there exists a neighborhood in C of the interval [0, (∆−1) ∆−1 (∆−2) ∆ ) on which the independence polynomial of any graph with maximum degree at most ∆ does not vanish. We show here that Sokal's Conjecture holds, as well as a multivariate version, and prove optimality for the domain of non-vanishing. An important step is to translate the setting to the language of complex dynamical systems.

We develop an efficient algorithmic approach for approximate counting and sampling in the low-temperature regime of a broad class of statistical physics models on finite subsets of the lattice Z d and on the torus (Z/nZ) d . Our approach is based on combining contour representations from Pirogov-Sinai theory with Barvinok's approach to approximate counting using truncated Taylor series. Some consequences of our main results include an FPTAS for approximating the partition function of the hard-core model at sufficiently high fugacity on subsets of Z d with appropriate boundary conditions and an efficient sampling algorithm for the ferromagnetic Potts model on the discrete torus (Z/nZ) d at sufficiently low temperature. 1 n log Z T d n (λ) of the hard-core model on Z d . That is, we would like an algorithm which for any ǫ > 0 outputs a number η ∈ [f d (λ) − ǫ, f d (λ) + ǫ] and whose running time grows as slowly as possible as a function of 1/ǫ. Gamarnik and Katz [28] gave such an algorithm running in time polynomial in 1/ǫ for λ small enough that the strong spatial mixing holds; this condition implies the hard-core model is in the uniqueness regime. Adams, Briceño, Marcus, and Pavlov [1] gave a polynomial-time algorithm for approximating the free energy of the hard-core (and several other) models on Z 2 in a subset of the uniqueness regime. More interestingly, their results also apply for the hard-core and Widom-Rowlinson models on Z 2 in a subset of the non-uniqueness regime. The latter result is of a similar spirit to the results of this paper.

Given complex numbers w 1 , . . . , w n , we define the weight w(X) of a set X of 0-1 vectors as the sum of w x 1 1 · · · w x n n over all vectors (x 1 , . . . , x n ) in X. We present an algorithm, which for a set X defined by a system of homogeneous linear equations with at most r variables per equation and at most c equations per variable, computes w(X) within relative error ǫ > 0 in (rc) O(ln n−ln ǫ) time provided |w j | ≤ β(r √ c) −1 for an absolute constant β > 0 and all j = 1, . . . , n. A similar algorithm is constructed for computing the weight of a linear code over F p . Applications include counting weighted perfect matchings in hypergraphs, counting weighted graph homomorphisms, computing weight enumerators of linear codes with sparse code generating matrices, and computing the partition functions of the ferromagnetic Potts model at low temperatures and of the hard-core model at high fugacity on biregular bipartite graphs.