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Exact Matching in Graphs of Bounded Independence Number

Authors Nicolas El Maalouly , Raphael Steiner



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Nicolas El Maalouly
  • Department of Computer Science, ETH Zürich, Switzerland
Raphael Steiner
  • Department of Computer Science, ETH Zürich, Switzerland

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Nicolas El Maalouly and Raphael Steiner. Exact Matching in Graphs of Bounded Independence Number. In 47th International Symposium on Mathematical Foundations of Computer Science (MFCS 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 241, pp. 46:1-46:14, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2022)
https://doi.org/10.4230/LIPIcs.MFCS.2022.46

Abstract

In the Exact Matching Problem (EM), we are given a graph equipped with a fixed coloring of its edges with two colors (red and blue), as well as a positive integer k. The task is then to decide whether the given graph contains a perfect matching exactly k of whose edges have color red. EM generalizes several important algorithmic problems such as perfect matching and restricted minimum weight spanning tree problems. When introducing the problem in 1982, Papadimitriou and Yannakakis conjectured EM to be NP-complete. Later however, Mulmuley et al. presented a randomized polynomial time algorithm for EM, which puts EM in RP. Given that to decide whether or not RP=P represents a big open challenge in complexity theory, this makes it unlikely for EM to be NP-complete, and in fact indicates the possibility of a deterministic polynomial time algorithm. EM remains one of the few natural combinatorial problems in RP which are not known to be contained in P, making it an interesting instance for testing the hypothesis RP=P. Despite EM being quite well-known, attempts to devise deterministic polynomial algorithms have remained illusive during the last 40 years and progress has been lacking even for very restrictive classes of input graphs. In this paper we push the frontier of positive results forward by proving that EM can be solved in deterministic polynomial time for input graphs of bounded independence number, and for bipartite input graphs of bounded bipartite independence number. This generalizes previous positive results for complete (bipartite) graphs which were the only known results for EM on dense graphs.

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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