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# Efficient Algorithms for Geometric Partial Matching

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## Acknowledgements

We thank Haim Kaplan for discussion and suggestion to use Goldberg et al. [Andrew V. Goldberg et al., 2017] algorithm. We thank Debmalya Panigrahi for helpful discussions.

## Cite As

Pankaj K. Agarwal, Hsien-Chih Chang, and Allen Xiao. Efficient Algorithms for Geometric Partial Matching. In 35th International Symposium on Computational Geometry (SoCG 2019). Leibniz International Proceedings in Informatics (LIPIcs), Volume 129, pp. 6:1-6:14, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2019)
https://doi.org/10.4230/LIPIcs.SoCG.2019.6

## Abstract

Let A and B be two point sets in the plane of sizes r and n respectively (assume r <= n), and let k be a parameter. A matching between A and B is a family of pairs in A x B so that any point of A cup B appears in at most one pair. Given two positive integers p and q, we define the cost of matching M to be c(M) = sum_{(a, b) in M}||a-b||_p^q where ||*||_p is the L_p-norm. The geometric partial matching problem asks to find the minimum-cost size-k matching between A and B. We present efficient algorithms for geometric partial matching problem that work for any powers of L_p-norm matching objective: An exact algorithm that runs in O((n + k^2)polylog n) time, and a (1 + epsilon)-approximation algorithm that runs in O((n + k sqrt{k})polylog n * log epsilon^{-1}) time. Both algorithms are based on the primal-dual flow augmentation scheme; the main improvements involve using dynamic data structures to achieve efficient flow augmentations. With similar techniques, we give an exact algorithm for the planar transportation problem running in O(min{n^2, rn^{3/2}}polylog n) time.

## Subject Classification

##### ACM Subject Classification
• Theory of computation → Design and analysis of algorithms
##### Keywords
• partial matching
• transportation
• optimal transport
• minimum-cost flow
• bichromatic closest pair

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## References

1. Pankaj K. Agarwal, Alon Efrat, and Micha Sharir. Vertical Decomposition of Shallow Levels in 3-Dimensional Arrangements and Its Applications. SIAM J. Comput., 29(3):912-953, 1999. URL: http://dx.doi.org/10.1137/S0097539795295936.
2. Pankaj K. Agarwal, Kyle Fox, Debmalya Panigrahi, Kasturi R. Varadarajan, and Allen Xiao. Faster Algorithms for the Geometric Transportation Problem. In Proc. 33rd Int. Sympos. Comput. Geom. (SoCG), pages 7:1-7:16, 2017. URL: http://dx.doi.org/10.4230/LIPIcs.SoCG.2017.7.
3. Pankaj K. Agarwal, Kyle Fox, Debmalya Panigrahi, Kasturi R. Varadarajan, and Allen Xiao. Faster Algorithms for the Geometric Transportation Problem. Preprint, 2019. URL: http://arxiv.org/abs/1903.08263.
4. Pankaj K. Agarwal and Kasturi R. Varadarajan. A near-linear constant-factor approximation for Euclidean bipartite matching? In Proc. 20th Annu. Sympos. Comput. Geom. (SoCG), pages 247-252, 2004. URL: http://dx.doi.org/10.1145/997817.997856.
5. D. Bertsekas and D. El Baz. Distributed Asynchronous Relaxation Methods for Convex Network Flow Problems. SIAM J. Control and Opt., 25(1):74-85, 1987. URL: http://dx.doi.org/10.1137/0325006.
6. Ran Duan, Seth Pettie, and Hsin-Hao Su. Scaling Algorithms for Weighted Matching in General Graphs. ACM Trans. Algorithms, 14(1):8:1-8:35, 2018. URL: http://dx.doi.org/10.1145/3155301.
7. Shimon Even and Robert E. Tarjan. Network Flow and Testing Graph Connectivity. SIAM J. Comput., 4(4):507-518, 1975. URL: http://dx.doi.org/10.1137/0204043.
8. Harold N. Gabow and Robert E. Tarjan. Faster Scaling Algorithms for Network Problems. SIAM J. Comput., 18(5):1013-1036, 1989. URL: http://dx.doi.org/10.1137/0218069.
9. Andrew V. Goldberg, Sagi Hed, Haim Kaplan, and Robert E. Tarjan. Minimum-Cost Flows in Unit-Capacity Networks. Theoret. Comput. Sci., 61(4):987-1010, 2017. URL: http://dx.doi.org/10.1007/s00224-017-9776-7.
10. John E. Hopcroft and Richard M. Karp. An n^5/2 Algorithm for Maximum Matchings in Bipartite Graphs. SIAM J. Comput., 2(4):225-231, 1973. URL: http://dx.doi.org/10.1137/0202019.
11. Haim Kaplan, Wolfgang Mulzer, Liam Roditty, Paul Seiferth, and Micha Sharir. Dynamic Planar Voronoi Diagrams for General Distance Functions and their Algorithmic Applications. In Proc. 28th Annu. ACM-SIAM Sympos. Discrete Algorithms (SODA), pages 2495-2504, 2017. URL: http://dx.doi.org/10.1137/1.9781611974782.165.
12. Andrey Boris Khesin, Aleksandar Nikolov, and Dmitry Paramonov. Preconditioning for the Geometric Transportation Problem. Preprint, 2019. URL: http://arxiv.org/abs/1902.08384.
13. Harold W. Kuhn. The Hungarian method for the assignment problem. Naval Research Logistics (NRL), 2(1-2):83-97, 1955.
14. Yin Tat Lee and Aaron Sidford. Path Finding Methods for Linear Programming: Solving Linear Programs in Õ(√rank) Iterations and Faster Algorithms for Maximum Flow. In 55th Annu. IEEE Sympos. Found. Comput. Sci. (FOCS), pages 424-433, 2014. URL: http://dx.doi.org/10.1109/FOCS.2014.52.
15. Aleksander Mądry. Navigating Central Path with Electrical Flows: From Flows to Matchings, and Back. In 54th Annu. IEEE Sympos. Found. Comput. Sci. (FOCS), pages 253-262, 2013. URL: http://dx.doi.org/10.1109/FOCS.2013.35.
16. James B. Orlin. A Faster Strongly Polynomial Minimum Cost Flow Algorithm. Operations Research, 41(2):338-350, 1993. URL: http://dx.doi.org/10.1287/opre.41.2.338.
17. Lyle Ramshaw and Robert E. Tarjan. A Weight-Scaling Algorithm for Min-Cost Imperfect Matchings in Bipartite Graphs. In Proc. 53rd Annu. IEEE Sympos. Found. Comput. Sci. (FOCS), pages 581-590, 2012. URL: http://dx.doi.org/10.1109/FOCS.2012.9.
18. R. Sharathkumar and Pankaj K. Agarwal. A near-linear time ε-approximation algorithm for geometric bipartite matching. In Proc. 44th Annu. ACM Sympos. Theory Comput. (STOC), pages 385-394, 2012. URL: http://dx.doi.org/10.1145/2213977.2214014.
19. R. Sharathkumar and Pankaj K. Agarwal. Algorithms for the transportation problem in geometric settings. In Proc. 23rd Annu. ACM-SIAM Sympos. Discrete Algorithms (SODA), pages 306-317, 2012. URL: https://dl.acm.org/citation.cfm?id=2095116.2095145.
20. Éva Tardos. A strongly polynomial minimum cost circulation algorithm. Combinatorica, 5(3):247-256, 1985. URL: http://dx.doi.org/10.1007/BF02579369.
21. Pravin M. Vaidya. Geometry Helps in Matching. SIAM J. Comput., 18(6):1201-1225, 1989. URL: http://dx.doi.org/10.1137/0218080.
22. Kasturi R. Varadarajan. A Divide-and-Conquer Algorithm for Min-Cost Perfect Matching in the Plane. In Proc. 39th Annu. IEEE Sympos. Found. Comput. Sci. (FOCS), pages 320-331, 1998. URL: http://dx.doi.org/10.1109/SFCS.1998.743466.
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