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Exact Matching: Correct Parity and FPT Parameterized by Independence Number

Authors Nicolas El Maalouly , Raphael Steiner , Lasse Wulf



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Author Details

Nicolas El Maalouly
  • Department of Computer Science, ETH Zürich, Switzerland
Raphael Steiner
  • Department of Computer Science, ETH Zürich, Switzerland
Lasse Wulf
  • Institute of Discrete Mathematics, TU Graz, Austria

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Nicolas El Maalouly, Raphael Steiner, and Lasse Wulf. Exact Matching: Correct Parity and FPT Parameterized by Independence Number. In 34th International Symposium on Algorithms and Computation (ISAAC 2023). Leibniz International Proceedings in Informatics (LIPIcs), Volume 283, pp. 28:1-28:18, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2023)
https://doi.org/10.4230/LIPIcs.ISAAC.2023.28

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.

Subject Classification

ACM Subject Classification
  • Theory of computation → Design and analysis of algorithms
  • Theory of computation → Parameterized complexity and exact algorithms
Keywords
  • Perfect Matching
  • Exact Matching
  • Independence Number
  • Parameterized Complexity

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