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Geometric Matching and Bottleneck Problems

Authors Sergio Cabello , Siu-Wing Cheng , Otfried Cheong , Christian Knauer

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  • Theory of computation → Computational geometry
Keywords
  • Many-to-many matching
  • bipartite
  • planar
  • geometric
  • approximation

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Abstract

Let P be a set of at most n points and let R be a set of at most n geometric ranges, such as disks and rectangles, where each p ∈ P has an associated supply s_{p} > 0, and each r ∈ R has an associated demand d_r > 0. A (many-to-many) matching is a set 𝒜 of ordered triples (p,r,a_{pr}) ∈ P × R × ℝ_{> 0} such that p ∈ r and the a_{pr}’s satisfy the constraints given by the supplies and demands. We show how to compute a maximum matching, that is, a matching maximizing ∑_{(p,r,a_{pr}) ∈ 𝒜} a_{pr}.
Using our techniques, we can also solve minimum bottleneck problems, such as computing a perfect matching between a set of n red points P and a set of n blue points Q that minimizes the length of the longest edge. For the L_∞-metric, we can do this in time O(n^{1+ε}) in any fixed dimension, for the L₂-metric in the plane in time O(n^{4/3 + ε}), for any ε > 0.

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Sergio Cabello, Siu-Wing Cheng, Otfried Cheong, and Christian Knauer. Geometric Matching and Bottleneck Problems. In 40th International Symposium on Computational Geometry (SoCG 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 293, pp. 31:1-31:15, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024) https://doi.org/10.4230/LIPIcs.SoCG.2024.31

Author Details

Sergio Cabello
  • University of Ljubljana, Ljubljana, Slovenia
  • Institute of Mathematics, Physics and Mechanics, Ljubljana, Slovenia
Siu-Wing Cheng
  • HKUST, Hong Kong, China
Otfried Cheong
  • SCALGO, Aarhus, Denmark
Christian Knauer
  • University of Bayreuth, Germany

Funding

This research was funded in part by the Slovenian Research and Innovation Agency (P1-0297, J1-2452, N1-0218, N1-0285), in part by the Research Grants Council, Hong Kong, China (project no. 16207419), and in part by the European Union (ERC, KARST, 101071836). Views and opinions expressed are however those of the authors only and do not necessarily reflect those of the European Union or the European Research Council. Neither the European Union nor the granting authority can be held responsible for them.

Acknowledgements

The authors thank Sang Won Bae for helpful discussions.

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