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

TL;DR

This paper introduces a fixed-parameter tractable algorithm for the exact matching problem in general graphs, extending previous work, and proves NP-hardness for a related relaxed problem.

Contribution

It presents a new deterministic FPT algorithm parameterized by the minimum odd cycle transversal and independence number, and establishes NP-hardness of a related problem.

Findings

Deterministic FPT algorithm for exact matching in general graphs.

Extension of previous bipartite graph results to general graphs.

NP-hardness of a relaxed parity matching problem.

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.