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Maximum Matchings in Geometric Intersection Graphs

Authors Édouard Bonnet , Sergio Cabello , Wolfgang Mulzer

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ACM Subject Classification
  • Theory of computation → Computational geometry
  • Theory of computation → Graph algorithms analysis
Keywords
  • computational geometry
  • geometric intersection graph
  • maximum matching
  • disk graph
  • unit-disk graph

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Abstract

Let G be an intersection graph of n geometric objects in the plane. We show that a maximum matching in G can be found in O(ρ^{3ω/2}n^{ω/2}) time with high probability, where ρ is the density of the geometric objects and ω>2 is a constant such that n × n matrices can be multiplied in O(n^ω) time.
The same result holds for any subgraph of G, as long as a geometric representation is at hand. For this, we combine algebraic methods, namely computing the rank of a matrix via Gaussian elimination, with the fact that geometric intersection graphs have small separators.
We also show that in many interesting cases, the maximum matching problem in a general geometric intersection graph can be reduced to the case of bounded density. In particular, a maximum matching in the intersection graph of any family of translates of a convex object in the plane can be found in O(n^{ω/2}) time with high probability, and a maximum matching in the intersection graph of a family of planar disks with radii in [1, Ψ] can be found in O(Ψ⁶log^11 n + Ψ^{12 ω} n^{ω/2}) time with high probability.

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Édouard Bonnet, Sergio Cabello, and Wolfgang Mulzer. Maximum Matchings in Geometric Intersection Graphs. In 37th International Symposium on Theoretical Aspects of Computer Science (STACS 2020). Leibniz International Proceedings in Informatics (LIPIcs), Volume 154, pp. 31:1-31:17, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2020) https://doi.org/10.4230/LIPIcs.STACS.2020.31

Author Details

Édouard Bonnet
  • Univ Lyon, CNRS, ENS de Lyon, Université Claude Bernard Lyon 1, LIP UMR5668, France
Sergio Cabello
  • Faculty of Mathematics and Physics, University of Ljubljana, and IMFM, Slovenia
Wolfgang Mulzer
  • Institut für Informatik, Freie Universität Berlin, Germany

Funding

  • Cabello, Sergio: Supported by the Slovenian Research Agency (P1-0297, J1-9109, J1-8130, J1-8155).
  • Mulzer, Wolfgang: Supported in part by ERC StG 757609.

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