For two point sets A, B ⊂ ℝ^d, with |A| = |B| = n and d > 1 a constant, and for a parameter ε > 0, we present a randomized algorithm that, with probability at least 1/2, computes in O(n(ε^{-O(d³)}log log n + ε^{-O(d)}log⁴ nlog⁵log n)) time, an ε-approximate minimum-cost perfect matching under any L_p-metric. All previous algorithms take n(ε^{-1}log n)^{Ω(d)} time. We use a randomly-shifted tree, with a polynomial branching factor and O(log log n) height, to define a tree-based distance function that ε-approximates the L_p metric as well as to compute the matching hierarchically. Then, we apply the primal-dual framework on a compressed representation of the residual graph to obtain an efficient implementation of the Hungarian-search and augment operations.
@InProceedings{agarwal_et_al:LIPIcs.SWAT.2022.6, author = {Agarwal, Pankaj K. and Raghvendra, Sharath and Shirzadian, Pouyan and Sowle, Rachita}, title = {{An Improved \epsilon-Approximation Algorithm for Geometric Bipartite Matching}}, booktitle = {18th Scandinavian Symposium and Workshops on Algorithm Theory (SWAT 2022)}, pages = {6:1--6:20}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-236-5}, ISSN = {1868-8969}, year = {2022}, volume = {227}, editor = {Czumaj, Artur and Xin, Qin}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.SWAT.2022.6}, URN = {urn:nbn:de:0030-drops-161660}, doi = {10.4230/LIPIcs.SWAT.2022.6}, annote = {Keywords: Euclidean bipartite matching, approximation algorithms, primal dual method} }
Feedback for Dagstuhl Publishing