2026-07-25T16:02:39Z https://drops.dagstuhl.de/oai

oai:drops-oai.dagstuhl.de:10410 2024-03-06T10:45:55Z ddc:004 open_access

Efficient Algorithms for Geometric Partial Matching Agarwal, Pankaj K. Chang, Hsien-Chih Xiao, Allen partial matching transportation optimal transport minimum-cost flow bichromatic closest pair 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. Schloss Dagstuhl – Leibniz-Zentrum für Informatik Pankaj K. Agarwal and Hsien-Chih Chang and Allen Xiao 2019 Is Part Of LIPIcs, Volume 129, 35th International Symposium on Computational Geometry (SoCG 2019) InProceedings Text doc-type:ResearchArticle publishedVersion application/pdf doi:10.4230/LIPIcs.SoCG.2019.6 urn:nbn:de:0030-drops-104109 https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.SoCG.2019.6 eng https://creativecommons.org/licenses/by/3.0/legalcode