# Weighted counting of solutions to sparse systems of equations

## TL;DR

This paper introduces an efficient algorithm for approximately counting weighted solutions to sparse linear systems and related combinatorial structures, with applications in statistical physics, coding theory, and graph theory.

## Contribution

It presents a novel algorithm for computing weighted counts in sparse systems with controlled error, extending to linear codes and various graph models.

## Key findings

- Algorithm computes weights within relative error efficiently
- Applicable to sparse linear systems and codes
- Enables approximate counting in complex combinatorial models

## Abstract

Given complex numbers $w_1, \ldots, w_n$, 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 $(x_1, \ldots, x_n)$ 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 $\epsilon >0$ in $(rc)^{O(\ln n-\ln \epsilon)}$ time provided $|w_j| \leq \beta (r \sqrt{c})^{-1}$ for an absolute constant $\beta >0$ and all $j=1, \ldots, n$. A similar algorithm is constructed for computing the weight of a linear code over ${\Bbb F}_p$. 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.