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

Authors Sayan Bandyapadhyay , Anil Maheshwari , Michiel Smid

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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


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

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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)


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
  • Many-to-many matching
  • bipartite
  • planar
  • geometric
  • approximation


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