| Title: | Geometric matching and bottleneck problems |
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| Authors: | ID Cabello, Sergio (Author)
ID Cheng, Siu-Wing (Author)
ID Cheong, Otfried (Author)
ID Knauer, Christian (Author) |
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| Files: |  PDF - Presentation file, download (959,05 KB)
MD5: BD886FA6FD5797E936ECDFECA6F0A6A0
 URL - Source URL, visit https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.SoCG.2024.31
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| Language: | English |
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| Typology: | 1.08 - Published Scientific Conference Contribution |
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| Organization: |  IMFM - Institute of Mathematics, Physics, and Mechanics
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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 or rectangles, where each $p \in P$ has an associated supply $s_{p} > 0$, and each $r \in R$ has an associated demand $d_{r} > 0$. A (many-to-many) matching is a set ${\mathcal A}$ of ordered triples $(p,r,a_{pr}) \in P \times R \times {\mathbb R}_{>0}$ such that $p \in r$ 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 $\sum_{(p,r,a_{pr}) \in {\mathcal A}} a_{pr}$. 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 $L_\infty$-metric, we can do this in time $O(n^{1+\varepsilon})$ in any fixed dimension, for the $L_2$-metric in the plane in time $O(n^{4/3 + \varepsilon})$, for any $\varepsilon > 0$. |
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| Keywords: | many-to-many matching, bipartite, planar, geometric matching, approximation |
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| Publication status: | Published |
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| Publication version: | Version of Record |
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| Year of publishing: | 2024 |
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| Number of pages: | Str. 31:1-31:15 |
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| PID: | 20.500.12556/DiRROS-20967  |
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| UDC: | 004.42 |
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| DOI: | 10.4230/LIPIcs.SoCG.2024.31 |
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| COBISS.SI-ID: | 218353667 |
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| Note: |
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| Publication date in DiRROS: | 11.12.2024 |
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| Views: | 59 |
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| Downloads: | 24 |
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| Metadata: |    |
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