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Weighted counting of solutions to sparse systems of equations

Part of: Discrete geometry Polytopes and polyhedra Equilibrium statistical mechanics Theory of computing Algorithms - Computer Science

Published online by Cambridge University Press:  15 April 2019

Alexander Barvinok  and
Guus Regts
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 author. Email: barvinok@umich.edu
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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.

MSC classification

Primary: 68Q25: Analysis of algorithms and problem complexity
Secondary: 68W25: Approximation algorithms 82B20: Lattice systems (Ising, dimer, Potts, etc.) and systems on graphs 52C07: Lattices and convex bodies in $n$ dimensions 52B55: Computational aspects related to convexity
Type
Paper
Information
Combinatorics, Probability and Computing , Volume 28 , Issue 5 , September 2019 , pp. 696 - 719
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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