Weighted counting of solutions to sparse systems of equations

Abstract

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.

A Deterministic Algorithm for Counting Colorings with 2-Delta Colors

“…1 These algorithms have made use of one of two main techniques: decay of correlations, which exploits decreasing in uence of the spins (colors) on distant vertices on the spin at a given vertex; and polynomial interpolation, which uses the absence of zeros of the partition function in a suitable region of the complex plane. Early examples of the decay of correlations approach include [1,2,40], while for early examples of the polynomial interpolation method, we refer to the monograph of Barvinok [3] (see also, e.g., [4,13,25,27,30,34] for more recent examples). Unfortunately, however, in the case of colorings on general bounded degree graphs, these techniques have so far lagged well behind the MCMC algorithms mentioned above.…”

Counting independent sets in unbalanced bipartite graphs

“…Here we give an FPTAS for the hard-core model on bipartite graphs of bounded degree whenever there is sufficient asymmetry in the degrees on either side or the fugacities assigned to the respective sides of the bipartition. In most biregular cases our results give significant improvement to the parameters from [2], as we discuss below, and our algorithm does not require biregularity. More importantly, the method, based on the cluster expansion and the Kotecký-Preiss condition [19] and related to that of [15,16], gives detailed probabilistic information about the hard-core model in addition to the algorithmic results.…”

“…Building on this approach, Jenssen, Keevash, and Perkins [16] gave an FPTAS for the hard-core model on bipartite expander graphs at large λ, and these results were sharpened in the case of random regular bipartite graphs [17,21]. Most relevant for this paper, Barvinok and Regts [2] give an FPTAS for Z(G) for biregular, bipartite graphs with unequal degrees when the fugacity is sufficiently large, as an application of a much more general approximate counting result.…”

Lee–Yang zeros and the complexity of the ferromagnetic Ising model on bounded-degree graphs

“…This result can be viewed as a derandomisation of the Markov chain-based randomised algorithm by Jerrum and Sinclair [26] for 𝜆 > 0 (see also [22,11]), solving a longstanding problem. 2 As noted in [35,Remark p. 290], the 'no-field' case |𝜆| = 1 is unclear, since, on the one hand, we have the algorithm by [26] for 𝜆 = 1 and, on the other hand, it is known that Lee-Yang zeros are dense on the unit circle. The density picture was further explored in [42] for graphs of bounded maximum degree Δ, by establishing for each 𝑏 ∈ (0, 1) a symmetric arc around 𝜆 = 1 on the unit circle where the partition function does not vanish for all graphs of maximum degree at most Δ and showing density of the Lee-Yang zeros on the complementary arc.…”

“…In particular, many of the recent advances on the development of approximation algorithms for counting problems have been based on viewing the partition function as a polynomial of the underlying parameters in the complex plane and using refined interpolation techniques from [1,39] to obtain fully polynomial time approximation schemes (FPTAS; see Subsection 1.1 below for the technical definition), even for real values [23,24,33,5,2,34,44,42,41]. The bottleneck of this approach is establishing zerofree regions in the complex plane of the polynomials, which in turn requires an in-depth understanding of the models with complex-valued parameters.…”