An analysis of a random algorithm for estimating all the matchings

TL;DR

This paper evaluates a simple approximation method for counting all matchings in bipartite graphs via the permanent of a matrix, revealing its limitations and failure to provide accurate estimates in most cases.

Contribution

It demonstrates that the Rasmussen method's critical ratio is too large for reliable counting of all matchings, highlighting the need for more effective estimation techniques.

Findings

The Rasmussen method's critical ratio is very large for most 0-1 matrices.

The method fails to accurately estimate the number of all matchings in bipartite graphs.

Estimation of the permanent for this purpose requires more careful approaches.

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.