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arXiv · 2405.02829

An FPT Algorithm for the Exact Matching Problem and NP-hardness of Related Problems

Hitoshi Murakami·Yutaro Yamaguchi
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Abstract

The exact matching problem is a constrained variant of the maximum matching problem: given a graph with each edge having a weight $0$ or $1$ and an integer $k$, the goal is to find a perfect matching of weight exactly $k$. Mulmuley, Vazirani, and Vazirani (1987) proposed a randomized polynomial-time algorithm for this problem, and it is still open whether it can be derandomized. Very recently, El Maalouly, Steiner, and Wulf (2023) showed that for bipartite graphs there exists a deterministic FPT algorithm parameterized by the (bipartite) independence number. In this paper, by extending a part of their work, we propose a deterministic FPT algorithm in general parameterized by the minimum size of an odd cycle transversal in addition to the (bipartite) independence number. We also consider a relaxed problem called the correct parity matching problem, and show that a slight generalization of an equivalent problem is NP-hard.

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Hitoshi Murakami, Yutaro Yamaguchi. 2024-05-05. An FPT Algorithm for the Exact Matching Problem and NP-hardness of Related Problems. https://arxiv.org/abs/2405.02829

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