Exact Matching: Correct Parity and FPT Parameterized by Independence Number

TL;DR

This paper advances the algorithmic understanding of the exact matching problem, providing an FPT algorithm for bipartite graphs and polynomial-time solutions for certain parity conditions in general graphs, addressing a longstanding open problem.

Contribution

It introduces novel fixed-parameter tractable algorithms for the exact matching problem based on independence number, improving upon previous XP algorithms and tackling the problem's complexity.

Findings

FPT algorithm for bipartite graphs with bounded independence number

Polynomial-time computation of perfect matchings with correct red edge parity in general graphs

Reduction of the general problem to finding perfect matchings with at most k red edges and correct parity

Abstract

Given an integer $k$ and a graph where every edge is colored either red or blue, the goal of the exact matching problem is to find a perfect matching with the property that exactly $k$ of its edges are red. Soon after Papadimitriou and Yannakakis (JACM 1982) introduced the problem, a randomized polynomial-time algorithm solving the problem was described by Mulmuley et al. (Combinatorica 1987). Despite a lot of effort, it is still not known today whether a deterministic polynomial-time algorithm exists. This makes the exact matching problem an important candidate to test the popular conjecture that the complexity classes P and RP are equal. In a recent article (MFCS 2022), progress was made towards this goal by showing that for bipartite graphs of bounded bipartite independence number, a polynomial time algorithm exists. In terms of parameterized complexity, this algorithm was an XP-algorithm parameterized by the bipartite independence number. In this article, we introduce novel algorithmic techniques that allow us to obtain an FPT-algorithm. If the input is a general graph we show that one can at least compute a perfect matching $M$ which has the correct number of red edges modulo 2, in polynomial time. This is motivated by our last result, in which we prove that an FPT algorithm for general graphs, parameterized by the independence number, reduces to the problem of finding in polynomial time a perfect matching $M$ with at most $k$ red edges and the correct number of red edges modulo 2.