Polynomial-Delay Enumeration of Large Maximal Matchings

TL;DR

This paper introduces efficient polynomial-delay algorithms for enumerating large maximal matchings in graphs, including variants that list matchings in non-decreasing order of size, advancing enumeration algorithm capabilities.

Contribution

It presents the first polynomial-delay algorithms for enumerating maximal matchings above a size threshold, including space-efficient and ordered variants.

Findings

Algorithms run in $O(nm)$ delay.

Existence of polynomial-space enumeration method.

Enumeration in non-decreasing order of size.

Abstract

Enumerating matchings is a classical problem in the field of enumeration algorithms. There are polynomial-delay enumeration algorithms for several settings, such as enumerating perfect matchings, maximal matchings, and (weighted) matchings in specific orders. In this paper, we present polynomial-delay enumeration algorithms for maximal matchings with cardinality at least given threshold $t$. Our algorithm enumerates all such matchings in $O(nm)$ delay with exponential space, where $n$ and $m$ are the number of vertices and edges of an input graph, respectively. We also present a polynomial-delay and polynomial-space enumeration algorithm for this problem. As a variant of this algorithm, we give an algorithm that enumerates all maximal matchings in non-decreasing order of its cardinality and runs in $O(nm)$ delay.