Document

# Geometric Matching and Bottleneck Problems

## File

LIPIcs.SoCG.2024.31.pdf
• Filesize: 0.93 MB
• 15 pages

## Acknowledgements

The authors thank Sang Won Bae for helpful discussions.

## Cite As

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

## 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.

## Subject Classification

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

## Metrics

• Access Statistics
• Total Accesses (updated on a weekly basis)
0

## References

1. Pankaj K. Agarwal, Noga Alon, Boris Aronov, and Subhash Suri. Can visibility graphs be represented compactly? Discrete & Computational Geometry, 12:347-365, 1994. URL: https://doi.org/10.1007/BF02574385.
2. Pankaj K. Agarwal and Jeff Erickson. Geometric range searching and its relatives. In B. Chazelle, J. Goodman, and R. Pollack, editors, Advances in Discrete and Computational Geometry, pages 1-56. AMS, 1998.
3. Pankaj K. Agarwal, Esther Ezra, and Micha Sharir. Semi-algebraic off-line range searching and biclique partitions in the plane. In 40th International Symposium on Computational Geometry, SoCG 2024, LIPIcs. Schloss Dagstuhl - Leibniz-Zentrum für Informatik, 2024.
4. Édouard Bonnet, Sergio Cabello, and Wolfgang Mulzer. Maximum matchings in geometric intersection graphs. Discrete & Computational Geometry, 70:550-579, 2023. URL: https://doi.org/10.1007/s00454-023-00564-3.
5. Rainer Burkard, Mauro Dell'Amico, and Silvano Martello. Assignment Problems. Society for Industrial and Applied Mathematics, 2012. URL: https://doi.org/10.1137/1.9781611972238.
6. Sergio Cabello, Siu-Wing Cheng, Otfried Cheong, and Christian Knauer. Geometric matching and bottleneck problems, 2023. URL: https://arxiv.org/abs/2310.02637.
7. John Gunnar Carlsson, Benjamin Armbruster, Saladi Rahul, and Haritha Bellam. A bottleneck matching problem with edge-crossing constraints. International Journal of Computational Geometry & Applications, 25:245-261, 2015. URL: https://doi.org/10.1142/S0218195915500144.
8. Bernard Chazelle, Herbert Edelsbrunner, Leonidas J. Guibas, and Micha Sharir. Algorithms for bichromatic line-segment problems and polyhedral terrains. Algorithmica, 11(2):116-132, 1994. URL: https://doi.org/10.1007/BF01182771.
9. Li Chen, Rasmus Kyng, Yang P. Liu, Richard Peng, Maximilian Probst Gutenberg, and Sushant Sachdeva. Maximum flow and minimum-cost flow in almost-linear time. In 63rd IEEE Annual Symposium on Foundations of Computer Science, FOCS 2022, Denver, CO, USA, October 31 - November 3, 2022, pages 612-623. IEEE, 2022. Full version at https://arxiv.org/abs/2203.00671. URL: https://doi.org/10.1109/FOCS54457.2022.00064.
10. David Cohen-Steiner, Herbert Edelsbrunner, and John Harer. Stability of persistence diagrams. Discret. Comput. Geom., 37(1):103-120, 2007. URL: https://doi.org/10.1007/s00454-006-1276-5.
11. Mark de Berg, Otfried Cheong, Marc van Kreveld, and Mark Overmars. Computational Geometry: Algorithms and Applications. Springer-Verlag, 3rd ed. edition, 2008. URL: https://doi.org/10.1007/978-3-540-77974-2.
12. Tamal Krishna Dey and Yusu Wang. Computational Topology for Data Analysis. Cambridge University Press, 2022. URL: https://doi.org/10.1017/9781009099950.
13. Herbert Edelsbrunner and John Harer. Computational Topology - an Introduction. American Mathematical Society, 2010. URL: http://www.ams.org/bookstore-getitem/item=MBK-69.
14. Alon Efrat, Alon Itai, and Matthew J. Katz. Geometry helps in bottleneck matching and related problems. Algorithmica, 31:1-28, 2001.
15. Ashish Goel, Michael Kapralov, and Sanjeev Khanna. Perfect matchings in O(n log n) time in regular bipartite graphs. SIAM J. Comput., 42(3):1392-1404, 2013. URL: https://doi.org/10.1137/100812513.
16. Sariel Har-Peled and Everett Yang. Approximation algorithms for maximum matchings in geometric intersection graphs. In 38th International Symposium on Computational Geometry, SoCG 2022, volume 224 of LIPIcs, pages 47:1-47:13. Schloss Dagstuhl - Leibniz-Zentrum für Informatik, 2022. URL: https://doi.org/10.4230/LIPIcs.SoCG.2022.47.
17. John E. Hopcroft and Richard M. Karp. An n^5/2 algorithm for maximum matchings in bipartite graphs. SIAM J. Comput., 2:225-231, 1973.
18. A. Karim Abu-Affash, Paz Carmi, Matthew J. Katz, and Yohai Trabelsi. Bottleneck non-crossing matching in the plane. Computational Geometry, 47:447-457, 2014. URL: https://doi.org/10.1016/j.comgeo.2013.10.005.
19. Matthew J. Katz and Micha Sharir. An expander-based approach to geometric optimization. SIAM J. Comput., 26:1384-1408, 1997. URL: https://doi.org/10.1137/S0097539794268649.
20. Matthew J. Katz and Micha Sharir. Bottleneck matching in the plane. Comput. Geom., 112:101986, 2023. URL: https://doi.org/10.1016/j.comgeo.2023.101986.
21. Michael Kerber, Dmitriy Morozov, and Arnur Nigmetov. Geometry helps to compare persistence diagrams. ACM J. Exp. Algorithmics, 22, 2017. URL: https://doi.org/10.1145/3064175.
22. Valerie King, S. Rao, and Robert Endre Tarjan. A faster deterministic maximum flow algorithm. J. Algorithms, 17(3):447-474, 1994. URL: https://doi.org/10.1006/jagm.1994.1044.
23. Jon Kleinberg and Éva Tardos. Algorithm Design. Addison-Wesley, 2005.
24. Cheng Long and Raymond Chi-Wing Wong. Optimal worst-case matching. In Shashi Shekhar, Hui Xiong, and Xun Zhou, editors, Encyclopedia of GIS, pages 1511-1522. Springer, 2017. URL: https://doi.org/10.1007/978-3-319-17885-1_1516.
25. Cheng Long, Raymond Chi-Wing Wong, Philip S. Yu, and Minhao Jiang. On optimal worst-case matching. In Proc. of the ACM SIGMOD Int. Conf. on Management of Data, pages 845-856. ACM, 2013.
26. Aleksander Mądry. Navigating central path with electrical flows: From flows to matchings, and back. In Proceedings of the 54th IEEE Annual Symposium on Foundations of Computer Science, FOCS 2013, pages 253-262, 2013. Full version at https://arxiv.org/abs/1307.2205. URL: https://doi.org/10.1109/FOCS.2013.35.
27. Jiří Matoušek. Range searching with efficient hierarchical cuttings. Discret. Comput. Geom., 10:157-182, 1993. URL: https://doi.org/10.1007/BF02573972.
28. Guillaume Moroz and Boris Aronov. Computing the distance between piecewise-linear bivariate functions. ACM Trans. Algorithms, 12(1):3:1-3:13, 2016. URL: https://doi.org/10.1145/2847257.
29. James B. Orlin. Max flows in O(nm) time, or better. In Dan Boneh, Tim Roughgarden, and Joan Feigenbaum, editors, Symposium on Theory of Computing Conference, STOC'13, Palo Alto, CA, USA, June 1-4, 2013, pages 765-774. ACM, 2013. URL: https://doi.org/10.1145/2488608.2488705.
30. James B. Orlin and Xiao-Yue Gong. A fast maximum flow algorithm. Networks, 77:287-321, 2021.
31. Daniel D. Sleator and Robert E. Tarjan. A data structure for dynamic trees. J. Comput. Syst. Sci., 26:362-391, 1983. URL: https://doi.org/10.1016/0022-0000(83)90006-5.
32. Robert E. Tarjan. Data Structures and Network Algorithms. Society for Industrial and Applied Mathematics, Philadelphia, PA, USA, 1983.
33. Leong Hou U, Man Lung Yiu, Kyriakos Mouratidis, and Nikos Mamoulis. Capacity constrained assignment in spatial databases. In Proc. of the ACM SIGMOD Int. Conf. on Management of Data, pages 15-28, 2008. URL: https://doi.org/10.1145/1376616.1376621.
34. Haitao Wang and Yiming Zhao. Improved algorithms for distance selection and related problems. In 31st Annual European Symposium on Algorithms, ESA 2023, volume 274 of LIPIcs, pages 101:1-101:14. Schloss Dagstuhl - Leibniz-Zentrum für Informatik, 2023. URL: https://doi.org/10.4230/LIPIcs.ESA.2023.101.
X

Feedback for Dagstuhl Publishing

### Thanks for your feedback!

Feedback submitted

### Could not send message

Please try again later or send an E-mail