Approximating the Permanent via Importance Sampling With Application to the Dimer Covering Problem

Published

February 1, 1999

Author(s)

Isabel M. Beichl, F Sullivan

Abstract

We estimate the asymptotic growth rate of the number of dimer covers of cubic lattice. Our estimate λ₃ = 0.4466 {plus or minus} 0.0006 is consistent with the lower bound obtained by Hammersley and the more recent improved upper bound obtained by Ciucu. It is well known that computing λ is equivalent ot computing the permanent of a certain 0 - 1 matrix. We describe an extremely efficient Monte Carlo algorithm for approximating the permanent. Previous work on Monte Carlo approaches includes the pioneering results of Jerrum and Sinclair, who use a rapidly mixing random walk. Our method is inspired by results of Soules on the convergence of Sinkhorn balancing. We use balancing to generate an importance function that allows us to do direct random sampling, rather than a random walk that converges to a limiting distribution.

Citation

Journal of Computational Physics

Volume

149

Issue

No. 1

Pub Type

Journals

Keywords

approximation methods, dimer covering problem, discrete mathematics, Monte Carlo, statistical physics

Citation

Beichl, I. and Sullivan, F. (1999), Approximating the Permanent via Importance Sampling With Application to the Dimer Covering Problem, Journal of Computational Physics (Accessed March 13, 2025)