# Exact Matching: Correct Parity and FPT Parameterized by Independence Number

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 introduced the problem in 1982, a randomized polynomial-time algorithm solving the problem was described by Mulmuley et al. 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, 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 independence number. In this article, we improve the techniques to obtain an FPT-algorithm for bipartite graphs. 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. This is motivated by our last result, in which we prove that an FPT algorithm for general graphs reduces to the problem of finding in polynomial time a perfect matching M with the correct number of red edges modulo 2 and additionally at most k red edges.

• 7 publications
• 18 publications
• 11 publications
research
07/05/2023

### An Approximation Algorithm for the Exact Matching Problem in Bipartite Graphs

In 1982 Papadimitriou and Yannakakis introduced the Exact Matching probl...
research
11/16/2022

### The Exact Bipartite Matching Polytope Has Exponential Extension Complexity

Given a graph with edges colored red or blue and an integer k, the exact...
research
02/24/2022

### Exact Matching in Graphs of Bounded Independence Number

In the Exact Matching Problem (EM), we are given a graph equipped with a...
research
03/25/2022

### Exact Matching: Algorithms and Related Problems

In 1982, Papadimitriou and Yannakakis introduced the Exact Matching (EM)...
research
09/17/2022

### Exact Matching and the Top-k Perfect Matching Problem

The aim of this note is to provide a reduction of the Exact Matching pro...
research
05/03/2018

### How to Secure Matchings Against Edge Failures

The matching preclusion number of a graph is the minimal number of edges...
research
10/22/2020

### An Investigation of the Recoverable Robust Assignment Problem

We investigate the so-called recoverable robust assignment problem on ba...