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Problem 6: Minimum Euclidean Matching in 2D


What is the complexity of computing a minimum-cost Euclidean matching for 2n points in the plane? The cost of a matching is the total length of the edges in the matching.


Uncertain, pending investigation.



Partial and Related Results

An algorithm that achieves the minimum and runs in nearly O(n^{2.5}) time has long been available [Vai89]. This was improved to O(n^{1.5} \log^5 n) in [Var98]. Recently Arora showed how to achieve a (1+\epsilon)-approximation in n (\log n)^{O(1/\epsilon)} time [Aro98], and this has been improved to O((n/\epsilon^3) \log^6 n) time [VA99].

A special case of considerable interest is bipartite matching, in which the points are red or blue and the matching connects points of different color. Here O(n^{2+\epsilon}) has been achieved [AES99], and a (1+\epsilon)-approximation can be found in O((n/\epsilon)^{1.5} \log^5 n) time [VA99].




shortest paths

Entry Revision History

J. O’Rourke, 2 Aug. 2001; 30 Aug. 2001; 13 Dec. 01 (thanks to M. Sharir).



Sanjeev Arora. Polynomial time approximation schemes for Euclidean traveling salesman and other geometric problems. J. Assoc. Comput. Mach., 45(5):753–782, 1998.


P. K. Agarwal, A. Efrat, and Micha Sharir. Vertical decomposition of shallow levels in 3-dimensional arrangements and its applications. SIAM J. Comput., 29:912–953, 1999.


J. S. B. Mitchell and Joseph O’Rourke. Computational geometry column 42. Internat. J. Comput. Geom. Appl., 11(5):573–582, 2001. Also in SIGACT News 32(3):63-72 (2001), Issue 120.


K. Varadarajan. A divide and conquer algorithm for min-cost perfect matching in the plane. In Proc. 39th Annu. IEEE Sympos. Found. Comput. Sci., pages 320–329, 1998.


P. M. Vaidya. Geometry helps in matching. SIAM J. Comput., 18:1201–1225, 1989.


K. R. Varadarajan and Pankaj K. Agarwal. Approximation algorithms for bipartite and non-bipartite matching in the plane. In Proc. 10th ACM-SIAM Sympos. Discrete Algorithms, pages 805–814, 1999.