Abstract
Let M be a perfect matching on a set of points in the plane where every edge is a line segment between two points. We say that M is globally maximum if it is a maximum-length matching on all points. We say that M is k-local maximum if for any subset \(M'=\{a_1b_1,\dots ,a_kb_k\}\) of k edges of M it holds that \(M'\) is a maximum-length matching on points \(\{a_1,b_1,\dots ,a_k,b_k\}\). We show that local maximum matchings are good approximations of global ones.
Let \(\mu _k\) be the infimum ratio of the length of any k-local maximum matching to the length of any global maximum matching, over all finite point sets in the Euclidean plane. It is known that \(\mu _k\geqslant \frac{k-1}{k}\) for any \(k\geqslant 2\). We show the following improved bounds for \(k\in \{2,3\}\): \(\mu _2\geqslant \sqrt{3/7} \) and \(\mu _3\geqslant 1/\sqrt{2}\). We also show that every pairwise crossing matching is unique and it is globally maximum.
Towards our proof of the lower bound for \(\mu _2\) we show the following result which is of independent interest: If we increase the radii of pairwise intersecting disks by factor \(2/\sqrt{3}\), then the resulting disks have a common intersection.
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This problem has applications in multicommodity flows in planar graphs [28].
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Biniaz, A., Maheshwari, A., Smid, M. (2021). Euclidean Maximum Matchings in the Plane—Local to Global. In: Lubiw, A., Salavatipour, M., He, M. (eds) Algorithms and Data Structures. WADS 2021. Lecture Notes in Computer Science(), vol 12808. Springer, Cham. https://doi.org/10.1007/978-3-030-83508-8_14
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