Weighted counting of solutions to sparse systems of equations. (15th September 2019)

• Record Type: Journal Article
• Authors: Barvinok, Alexander; Regts, Guus
• Is Part Of: Combinatorics, probability and computing. Volume 28:Number 5(2019)
• Page Start: 696
• Page End: 719
• Publication Date: 2019-09-15
• Subjects: Primary 68Q25, -- Secondary 68W25, -- 82B20, -- 52C07, -- 52B55; Combinatorial analysis -- Periodicals; Probabilities -- Periodicals; Computer science -- Mathematics -- Periodicals; 511.6
• DOI: 10.1017/S0963548319000105
• Languages: English
• ISSNs: 0963-5483

Abstract

Abstract: Given complex numbers w 1, …, wn, we define the weight w ( X ) of a set X of 0–1 vectors as the sum of $w_1^{x_1} \cdots w_n^{x_n}$ over all vectors ( x 1, …, xn ) 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 ∊ > 0 in ( rc ) O (ln n -ln ∊ ) time provided $|w_j| \leq \beta (r \sqrt{c})^{-1}$ 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 ${\mathbb F}_p$ . 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.