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Exact Matching: Algorithms and Related Problems

Nicolas ElMaalouly
In 1982, Papadimitriou and Yannakakis introduced the Exact Matching (EM)problem where given an edge colored graph, with colors red and blue, and aninteger $k$, the goal is to decided whether or not the graph contains a perfectmatching with exactly $k$ red edges. Although they conjectured it to be$\textbf{NP}$-complete, soon after it was shown to be solvable in randomizedpolynomial time in the seminal work of Mulmuley et al. placing it in thecomplexity class $\textbf{RP}$. Since then, all attempts at finding adeterministic algorithm for EM have failed, thus leaving it as one of the fewnatural combinatorial problems in $\textbf{RP}$ but not known to be containedin $\textbf{P}$, and making it an interesting instance for testing thehypothesis $\textbf{RP}=\textbf{P}$. Progress has been lacking even on veryrestrictive classes of graphs despite the problem being quite well known asevidenced by the number of works citing it. In this paper we aim to gain moreinsight into the problem by considering two directions of study. In the firstdirection, we study EM on bipartite graphs with a relaxation of the colorconstraint and provide an algorithm where the output is required to be aperfect 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 anoptimization problem, it is more natural to approximate so we provideapproximation algorithms for it. In the second direction, we look at theparameterized algorithms. Here we introduce new tools and FPT algorithms forthe study of EM and TkPM.