Parameterized and Approximation Algorithms for Finding Two Disjoint Matchings

Abstract

We first present a fixed-parameter algorithm for the NP-hard problem of deciding if there are two matchings M₁ and M₂ in a given graph G such that |M₁| + |M₂| is no less than a given number k. The algorithm runs in $O\left(m + k\cdot k!\cdot \left(2\sqrt{2}\right)^k\cdot n^2\log n \ \right)$ time, where n (respectively, m) is the number of vertices (respectively, edges) in G. We then present a combinatorial approximation algorithm for the NP-hard problem of finding two disjoint matchings in a given edge-weighted graph G so that their total weight is maximized. The algorithm achieves an approximation ratio of roughly 0.76 and runs in $O\left(m + n^3\alpha(n)\right)$ time, where α is the inverse Ackermann function.