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Weighted counting of solutions to sparse systems of equations

Published online by Cambridge University Press:  15 April 2019

Alexander Barvinok
Affiliation:
Department of Mathematics, University of Michigan, Ann Arbor, MI 48109-1043, USA
Guus Regts
Affiliation:
Korteweg de Vries Institute for Mathematics, University of Amsterdam, P.O. Box 94248, 1090 GE Amsterdam, The Netherlands
Corresponding
E-mail address:

Abstract

Given complex numbers w1,…,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 (x1,…,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(lnn-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.

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Paper
Copyright
© Cambridge University Press 2019 

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Footnotes

Research partially supported by NSF grant DMS 1361541.

Research supported by a personal NWO Veni grant.

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