eng
Schloss Dagstuhl β Leibniz-Zentrum fΓΌr Informatik
Leibniz International Proceedings in Informatics
1868-8969
2024-06-06
31:1
31:15
10.4230/LIPIcs.SoCG.2024.31
article
Geometric Matching and Bottleneck Problems
Cabello, Sergio
1
2
https://orcid.org/0000-0002-3183-4126
Cheng, Siu-Wing
3
https://orcid.org/0000-0002-3557-9935
Cheong, Otfried
4
https://orcid.org/0000-0003-4467-7075
Knauer, Christian
5
University of Ljubljana, Ljubljana, Slovenia
Institute of Mathematics, Physics and Mechanics, Ljubljana, Slovenia
HKUST, Hong Kong, China
SCALGO, Aarhus, Denmark
University of Bayreuth, Germany
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 β P has an associated supply s_{p} > 0, and each r β R has an associated demand d_r > 0. A (many-to-many) matching is a set π of ordered triples (p,r,a_{pr}) β P Γ R Γ β_{> 0} such that p β r and the a_{pr}βs satisfy the constraints given by the supplies and demands. We show how to compute a maximum matching, that is, a matching maximizing β_{(p,r,a_{pr}) β π} 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_β-metric, we can do this in time O(n^{1+Ξ΅}) in any fixed dimension, for the Lβ-metric in the plane in time O(n^{4/3 + Ξ΅}), for any Ξ΅ > 0.
https://drops.dagstuhl.de/storage/00lipics/lipics-vol293-socg2024/LIPIcs.SoCG.2024.31/LIPIcs.SoCG.2024.31.pdf
Many-to-many matching
bipartite
planar
geometric
approximation