Complexity of a Disjoint Matching Problem on Bipartite Graphs

TL;DR

This paper investigates the computational complexity of a disjoint matching problem in bipartite graphs, showing it is solvable in special cases but NP-hard in general.

Contribution

It establishes the NP-hardness of finding two disjoint matchings with specific saturation conditions in bipartite graphs when the subset S is arbitrary.

Findings

NP-hardness for general S

Polynomial solvability when |S| ≥ |X|-1

Complexity boundary for the problem

Abstract

We consider the following question: given an $(X,Y)$-bigraph $G$ and a set $S \subset X$, does $G$ contain two disjoint matchings $M_1$ and $M_2$ such that $M_1$ saturates $X$ and $M_2$ saturates $S$? When $|S|\geq |X|-1$, this question is solvable by finding an appropriate factor of the graph. In contrast, we show that when $S$ is allowed to be an arbitrary subset of $X$, the problem is NP-hard.