Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: 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.

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The categories of this paper are: comp-gas
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The categories of this paper are: q-bio
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The categories of this paper are: gr-qc
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The categories of this paper are: nlin
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The categories of this paper are: ao-sci
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The categories of this paper are: math
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The categories of this paper are: eess
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The categories of this paper are: q-alg
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The categories of this paper are: cond-mat
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The categories of this paper are: astro-ph
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The categories of this paper are: nucl-th
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The categories of this paper are: hep-ex
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The categories of this paper are: q-fin
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The categories of this paper are: hep-lat
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The categories of this paper are: quant-ph
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The categories of this paper are: physics