Diverse Pairs of Matchings

TL;DR

This paper studies the computational complexity of finding two matchings in a graph with a large symmetric difference, providing polynomial algorithms for bipartite graphs, fixed-parameter tractability results, and kernelization bounds.

Contribution

It introduces the Diverse Pair of Matchings problems, analyzes their complexity, and offers new algorithms and kernelization results for restricted graph classes.

Findings

Polynomial-time solution for bipartite graphs.

FPT algorithm for Diverse Pair of Maximum Matchings parameterized by k.

Kernel of size O(k^2) vertices for the problem.

Abstract

We initiate the study of the Diverse Pair of (Maximum/ Perfect) Matchings problems which given a graph $G$ and an integer $k$, ask whether $G$ has two (maximum/perfect) matchings whose symmetric difference is at least $k$. Diverse Pair of Matchings (asking for two not necessarily maximum or perfect matchings) is NP-complete on general graphs if $k$ is part of the input, and we consider two restricted variants. First, we show that on bipartite graphs, the problem is polynomial-time solvable, and second we show that Diverse Pair of Maximum Matchings is FPT parameterized by $k$. We round off the work by showing that Diverse Pair of Matchings has a kernel on $\mathcal{O}(k^2)$ vertices.