A New Direction for Counting Perfect Matchings

TL;DR

This paper introduces a novel exact algorithm for counting perfect matchings in bipartite graphs that improves time complexity by avoiding traditional methods like inclusion-exclusion and tree-decompositions, using a reduction to cut-weight distribution.

Contribution

The paper presents a new algorithm that leverages a reduction to cut-weight distribution and MacWilliams Identity, offering better scalability for graphs with higher degree.

Findings

Achieves faster time bounds for counting perfect matchings in bipartite graphs.

Introduces a new reduction method based on coding theory concepts.

Demonstrates improved performance over previous algorithms.

Abstract

In this paper, we present a new exact algorithm for counting perfect matchings, which relies on neither inclusion-exclusion principle nor tree-decompositions. For any bipartite graph of $2n$ nodes and $\Delta n$ edges such that $\Delta \geq 3$, our algorithm runs with $O^{\ast}(2^{(1 - 1/O(\Delta \log \Delta))n})$ time and exponential space. Compared to the previous algorithms, it achieves a better time bound in the sense that the performance degradation to the increase of $\Delta$ is quite slower. The main idea of our algorithm is a new reduction to the problem of computing the cut-weight distribution of the input graph. The primary ingredient of this reduction is MacWilliams Identity derived from elementary coding theory. The whole of our algorithm is designed by combining that reduction with a non-trivial fast algorithm computing the cut-weight distribution. To the best of our knowledge, the approach posed in this paper is new and may be of independent interest.