An analysis of a random algorithm for estimating all the matchings

TL;DR

This paper analyzes the performance of Rasmussen's simple approximation method for counting all matchings in bipartite graphs, showing it often fails due to large critical ratios, thus requiring more careful estimation techniques.

Contribution

It demonstrates that Rasmussen's method performs poorly for counting all matchings, highlighting the need for improved permanent estimation approaches in this context.

Findings

Critical ratio is very large with high probability.

RM fails to accurately estimate the number of all matchings.

Performance degrades as graph size increases.

Abstract

Counting the number of all the matchings on a bipartite graph has been transformed into calculating the permanent of a matrix obtained from the extended bipartite graph by Yan Huo, and Rasmussen presents a simple approach (RM) to approximate the permanent, which just yields a critical ratio O($n\omega(n)$) for almost all the 0-1 matrices, provided it's a simple promising practical way to compute this #P-complete problem. In this paper, the performance of this method will be shown when it's applied to compute all the matchings based on that transformation. The critical ratio will be proved to be very large with a certain probability, owning an increasing factor larger than any polynomial of $n$ even in the sense for almost all the 0-1 matrices. Hence, RM fails to work well when counting all the matchings via computing the permanent of the matrix. In other words, we must carefully utilize the known methods of estimating the permanent to count all the matchings through that transformation.