Tverberg’s theorem states that a set of n points in ℝ^d can be partitioned into ⌈n/(d+1)⌉ sets whose convex hulls all intersect. A point in the intersection (aka Tverberg point) is a centerpoint, or high-dimensional median, of the input point set. While randomized algorithms exist to find centerpoints with some failure probability, a partition for a Tverberg point provides a certificate of its correctness. Unfortunately, known algorithms for computing exact Tverberg points take n^{O(d²)} time. We provide several new approximation algorithms for this problem, which improve running time or approximation quality over previous work. In particular, we provide the first strongly polynomial (in both n and d) approximation algorithm for finding a Tverberg point.
@InProceedings{harpeled_et_al:LIPIcs.ESA.2021.51, author = {Har-Peled, Sariel and Zhou, Timothy}, title = {{Improved Approximation Algorithms for Tverberg Partitions}}, booktitle = {29th Annual European Symposium on Algorithms (ESA 2021)}, pages = {51:1--51:15}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-204-4}, ISSN = {1868-8969}, year = {2021}, volume = {204}, editor = {Mutzel, Petra and Pagh, Rasmus and Herman, Grzegorz}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ESA.2021.51}, URN = {urn:nbn:de:0030-drops-146323}, doi = {10.4230/LIPIcs.ESA.2021.51}, annote = {Keywords: Geometric spanners, vertex failures, robustness} }
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