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Exact and Approximation Algorithms for Many-To-Many Point Matching in the Plane

Authors Sayan Bandyapadhyay , Anil Maheshwari , Michiel Smid

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Author Details

Sayan Bandyapadhyay
  • Department of Informatics, University of Bergen, Norway
Anil Maheshwari
  • School of Computer Science, Carleton University, Ottawa, Canada
Michiel Smid
  • School of Computer Science, Carleton University, Ottawa, Canada

Funding

This work is supported by the European Research Council (ERC) via grant LOPPRE, reference 819416 and by NSERC, Canada.

Acknowledgements

We thank Saeed Mehrabi for introducing the many-to-many matching problem to us.

Cite AsGet BibTex

Sayan Bandyapadhyay, Anil Maheshwari, and Michiel Smid. Exact and Approximation Algorithms for Many-To-Many Point Matching in the Plane. In 32nd International Symposium on Algorithms and Computation (ISAAC 2021). Leibniz International Proceedings in Informatics (LIPIcs), Volume 212, pp. 44:1-44:14, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2021)
https://doi.org/10.4230/LIPIcs.ISAAC.2021.44

Abstract

Given two sets S and T of points in the plane, of total size n, a many-to-many matching between S and T is a set of pairs (p,q) such that p ∈ S, q ∈ T and for each r ∈ S ∪ T, r appears in at least one such pair. The cost of a pair (p,q) is the (Euclidean) distance between p and q. In the minimum-cost many-to-many matching problem, the goal is to compute a many-to-many matching such that the sum of the costs of the pairs is minimized. This problem is a restricted version of minimum-weight edge cover in a bipartite graph, and hence can be solved in O(n³) time. In a more restricted setting where all the points are on a line, the problem can be solved in O(nlog n) time [Justin Colannino et al., 2007]. However, no progress has been made in the general planar case in improving the cubic time bound. In this paper, we obtain an O(n²⋅ poly(log n)) time exact algorithm and an O(n^{3/2}⋅ poly(log n)) time (1+ε)-approximation in the planar case.

Subject Classification

ACM Subject Classification
  • Theory of computation → Approximation algorithms analysis
Keywords
  • Many-to-many matching
  • bipartite
  • planar
  • geometric
  • approximation

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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. Google Scholar
  2. Mudabir Kabir Asathulla, Sanjeev Khanna, Nathaniel Lahn, and Sharath Raghvendra. A faster algorithm for minimum-cost bipartite perfect matching in planar graphs. ACM Trans. Algorithms, 16(1):2:1-2:30, 2020. Google Scholar
  3. Justin Colannino, Mirela Damian, Ferran Hurtado, Stefan Langerman, Henk Meijer, Suneeta Ramaswami, Diane L. Souvaine, and Godfried Toussaint. Efficient many-to-many point matching in one dimension. Graphs Comb., 23(Supplement-1):169-178, 2007. URL: https://doi.org/10.1007/s00373-007-0714-3.
  4. Thomas Eiter and Heikki Mannila. Distance measures for point sets and their computation. Acta informatica, 34(2):109-133, 1997. Google Scholar
  5. S. M. Ferdous, Alex Pothen, and Arif Khan. New approximation algorithms for minimum weighted edge cover. In Fredrik Manne, Peter Sanders, and Sivan Toledo, editors, Proceedings of the Eighth SIAM Workshop on Combinatorial Scientific Computing, CSC 2018, Bergen, Norway, June 6-8, 2018, pages 97-108. SIAM, 2018. URL: https://doi.org/10.1137/1.9781611975215.10.
  6. Harold N Gabow and Robert E Tarjan. Faster scaling algorithms for network problems. SIAM Journal on Computing, 18(5):1013-1036, 1989. Google Scholar
  7. John E Hopcroft and Richard M Karp. An n^5/2 algorithm for maximum matchings in bipartite graphs. SIAM Journal on computing, 2(4):225-231, 1973. Google Scholar
  8. Haim Kaplan, Wolfgang Mulzer, Liam Roditty, Paul Seiferth, and Micha Sharir. Dynamic planar Voronoi diagrams for general distance functions and their algorithmic applications. Discrete & Computational Geometry, 64(3):838-904, 2020. Google Scholar
  9. Judith Keijsper and Rudi Pendavingh. An efficient algorithm for minimum-weight bibranching. Journal of Combinatorial Theory, Series B, 73(2):130-145, 1998. Google Scholar
  10. Harold W Kuhn. Variants of the Hungarian method for assignment problems. Naval Research Logistics Quarterly, 3(4):253-258, 1956. Google Scholar
  11. Sharath Raghvendra and Pankaj K. Agarwal. A near-linear time ε-approximation algorithm for geometric bipartite matching. J. ACM, 67(3):18:1-18:19, 2020. Google Scholar
  12. Fatemeh Rajabi-Alni and Alireza Bagheri. An O(n²) algorithm for the limited-capacity many-to-many point matching in one dimension. Algorithmica, 76(2):381-400, 2016. URL: https://doi.org/10.1007/s00453-015-0044-4.
  13. Fatemeh Rajabi-Alni and Alireza Bagheri. A fast and efficient algorithm for many-to-many matching of points with demands in one dimension. CoRR, abs/1904.05184, 2019. URL: http://arxiv.org/abs/1904.05184.
  14. Fatemeh Rajabi-Alni and Alireza Bagheri. A faster algorithm for the limited-capacity many-to-many point matching in one dimension. CoRR, abs/1904.03015, 2019. URL: http://arxiv.org/abs/1904.03015.
  15. A. Schrijver. Combinatorial Optimization - Polyhedra and Efficiency. Springer, 2003. Google Scholar
  16. Pravin M Vaidya. Geometry helps in matching. SIAM Journal on Computing, 18(6):1201-1225, 1989. Google Scholar
  17. Kasturi R Varadarajan and Pankaj K Agarwal. Approximation algorithms for bipartite and non-bipartite matching in the plane. In SODA, volume 99, pages 805-814, 1999. Google Scholar
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