Counting Perfect Matchings as Fast as Ryser

TL;DR

This paper introduces a polynomial space algorithm for counting perfect matchings in graphs with a runtime comparable to Ryser's bipartite algorithm, significantly improving previous methods and extending to hafnian computation.

Contribution

The authors present a new polynomial space algorithm that counts perfect matchings in $O^*(2^{n/2})$ time, matching Ryser's efficiency for bipartite graphs and generalizing to hafnian computation.

Findings

Achieves $O^*(2^{n/2})$ time complexity for counting perfect matchings

Matches the runtime of Ryser's algorithm for bipartite graphs

Shows limitations for faster algorithms for general set cover counting

Abstract

We show that there is a polynomial space algorithm that counts the number of perfect matchings in an $n$-vertex graph in $O^*(2^{n/2})\subset O(1.415^n)$ time. ($O^*(f(n))$ suppresses functions polylogarithmic in $f(n)$).The previously fastest algorithms for the problem was the exponential space $O^*(((1+\sqrt{5})/2)^n) \subset O(1.619^n)$ time algorithm by Koivisto, and for polynomial space, the $O(1.942^n)$ time algorithm by Nederlof. Our new algorithm's runtime matches up to polynomial factors that of Ryser's 1963 algorithm for bipartite graphs. We present our algorithm in the more general setting of computing the hafnian over an arbitrary ring, analogously to Ryser's algorithm for permanent computation. We also give a simple argument why the general exact set cover counting problem over a slightly superpolynomial sized family of subsets of an $n$ element ground set cannot be solved in $O^*(2^{(1-\epsilon_1)n})$ time for any $\epsilon_1>0$ unless there are $O^*(2^{(1-\epsilon_2)n})$ time algorithms for computing an $n\times n$ 0/1 matrix permanent, for some $\epsilon_2>0$ depending only on $\epsilon_1$.