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An Approximation Algorithm for the Exact Matching Problem in Bipartite Graphs

Authors Anita Dürr , Nicolas El Maalouly , Lasse Wulf

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LIPIcs.APPROX-RANDOM.2023.18.pdf
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ACM Subject Classification
  • Theory of computation → Design and analysis of algorithms
  • Theory of computation → Approximation algorithms analysis
Keywords
  • Perfect Matching
  • Exact Matching
  • Red-Blue Matching
  • Approximation Algorithms
  • Bounded Color Matching

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Abstract

In 1982 Papadimitriou and Yannakakis introduced the Exact Matching problem, in which given a red and blue edge-colored graph G and an integer k one has to decide whether there exists a perfect matching in G with exactly k red edges. Even though a randomized polynomial-time algorithm for this problem was quickly found a few years later, it is still unknown today whether a deterministic polynomial-time algorithm exists. This makes the Exact Matching problem an important candidate to test the RP=P hypothesis.
In this paper we focus on approximating Exact Matching. While there exists a simple algorithm that computes in deterministic polynomial-time an almost perfect matching with exactly k red edges, not a lot of work focuses on computing perfect matchings with almost k red edges. In fact such an algorithm for bipartite graphs running in deterministic polynomial-time was published only recently (STACS'23). It outputs a perfect matching with k' red edges with the guarantee that 0.5k ≤ k' ≤ 1.5k. In the present paper we aim at approximating the number of red edges without exceeding the limit of k red edges. We construct a deterministic polynomial-time algorithm, which on bipartite graphs computes a perfect matching with k' red edges such that k/3 ≤ k' ≤ k.

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Anita Dürr, Nicolas El Maalouly, and Lasse Wulf. An Approximation Algorithm for the Exact Matching Problem in Bipartite Graphs. In Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2023). Leibniz International Proceedings in Informatics (LIPIcs), Volume 275, pp. 18:1-18:21, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2023) https://doi.org/10.4230/LIPIcs.APPROX/RANDOM.2023.18

Author Details

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

Funding

  • Wulf, Lasse: Supported by the Austrian Science Fund (FWF): W1230.

References

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