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Article Dans Une Revue Discrete and Computational Geometry Année : 2023

Maximum Matchings in Geometric Intersection Graphs

Wolfgang Mulzer

Résumé

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, ] can be found in O (6 log 11 n + 12ω n ω/2) time with high probability.
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Dates et versions

hal-04296712 , version 1 (22-05-2024)

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Edouard Bonnet, Sergio Cabello, Wolfgang Mulzer. Maximum Matchings in Geometric Intersection Graphs. Discrete and Computational Geometry, 2023, 70, pp.550 - 579. ⟨10.1007/s00454-023-00564-3⟩. ⟨hal-04296712⟩
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