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

TL;DR

The paper advances the deterministic tractability of the exact matching problem (EM) in 0/1-weighted graphs by giving a fixed-parameter tractable (FPT) algorithm parameterized by the minimum odd cycle transversal size together with the (bipartite) independence number. The core approach extends prior bipartite results by reducing EM to the bounded correct parity matching problem (BCPM) on carefully constructed subgraphs $G_Y$ derived from an odd cycle transversal $X$, and solving these subproblems within FPT time. It also investigates the parity-constrained variants, showing a graph-theoretic equivalence between CPM and an odd alternating cycle problem (OAC), and proves the NP-hardness of a localized version (OACe), highlighting the need for global search strategies. Additionally, the paper provides heuristic speeding-up techniques and a detailed NP-hardness discussion for related problems, including reductions from back-and-forth path structures to establish computational hardness under various restrictions, while identifying a polynomial-time solvable variant (FDAP). Overall, the work broadens the algorithmic toolkit for EM by connecting structural graph parameters with parity-constrained matching, offering practical FPT algorithms and illuminating the landscape of related computational problems.

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.

Theorems & Definitions (24)

• Theorem 1.1: El Maalouly et al. ESW2023
• Theorem 1.2: El Maalouly et al. ESW2023
• Theorem 1.3: El Maalouly et al. ESW2023
• Theorem 1.4
• Theorem 1.5
• Lemma 3.1
• proof
• Lemma 3.2
• proof
• Remark 3.3