Exact Matching: Algorithms and Related Problems

In 1982, Papadimitriou and Yannakakis introduced the Exact Matching (EM) problem where given an edge colored graph, with colors red and blue, and an integer $k$, the goal is to decided whether or not the graph contains a perfect matching with exactly $k$ red edges. Although they conjectured it to be $\textbf{NP}$-complete, soon after it was shown to be solvable in randomized polynomial time in the seminal work of Mulmuley et al. placing it in the complexity class $\textbf{RP}$. Since then, all attempts at finding a deterministic algorithm for EM have failed, thus leaving it as one of the few natural combinatorial problems in $\textbf{RP}$ but not known to be contained in $\textbf{P}$, and making it an interesting instance for testing the hypothesis $\textbf{RP}=\textbf{P}$. Progress has been lacking even on very restrictive classes of graphs despite the problem being quite well known as evidenced by the number of works citing it. In this paper we aim to gain more insight into the problem by considering two directions of study. In the first direction, we study EM on bipartite graphs with a relaxation of the color constraint and provide an algorithm where the output is required to be a perfect matching with a number of red edges differing from $k$ by at most $k/2$. We also introduce an optimisation problem we call Top-k Perfect Matching (TkPM) that shares many similarities with EM. By virtue of being an optimization problem, it is more natural to approximate so we provide approximation algorithms for it. In the second direction, we look at the parameterized algorithms. Here we introduce new tools and FPT algorithms for the study of EM and TkPM.