Weighted counting of integer points in a subspace

Given complex numbers w1, . . . , wn, we define the weight w(X) of a set X of non-negative integer n-vectors as the sum of w1 1 · · ·w xn n over all vectors (x1, . . . , xn) in X. We present an algorithm, which for a set X of 0-1 vectors 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 a relative error ǫ > 0 in (rc)O(lnn−ln ǫ) time provided |wj | ≤ 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 set of non-negative integer vectors satisfying linear constraints and the weight of a linear code over Fp. Applications include counting of weighted perfect matchings in hypergraphs, computing permanents, counting weighted graph homomorphisms and computing weight enumerators of linear codes with sparse code generating matrices. 1. Weighted counting of integer points (1.1) Weight of a set of integer points. Let Z+ denote the set of non-negative integers. Let us fix complex numbers w1, . . . , wn, referred to as weights in what follows. We define the weight w(x) of a vector x ∈ Z+ by w(x) = w1 1 · · ·wn n where x = (ξ1, . . . , ξn) . Here we agree that 0 = 1, so that w(x) is a continuous function of w1, . . . , wn for a fixed x. We define the weight of a finite set X ⊂ Z+ by