We consider the problem of finding a small hitting set in an infinite range space F=(Q,R) of bounded VC-dimension. We show that, under reasonably general assumptions, the infinite-dimensional convex relaxation can be solved (approximately) efficiently by multiplicative weight updates. As a consequence, we get an algorithm that finds, for any delta>0, a set of size O(s_F(z^*_F)) that hits (1-delta)-fraction of R (with respect to a given measure) in time proportional to log(1/delta), where s_F(1/epsilon) is the size of the smallest epsilon-net the range space admits, and z^*_F is the value of the fractional optimal solution. This exponentially improves upon previous results which achieve the same approximation guarantees with running time proportional to poly(1/delta). Our assumptions hold, for instance, in the case when the range space represents the visibility regions of a polygon in the plane, giving thus a deterministic polynomial-time O(log z^*_F)-approximation algorithm for guarding (1-delta)-fraction of the area of any given simple polygon, with running time proportional to polylog(1/delta).
@InProceedings{elbassioni:LIPIcs.SoCG.2017.40, author = {Elbassioni, Khaled}, title = {{Finding Small Hitting Sets in Infinite Range Spaces of Bounded VC-Dimension}}, booktitle = {33rd International Symposium on Computational Geometry (SoCG 2017)}, pages = {40:1--40:15}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-038-5}, ISSN = {1868-8969}, year = {2017}, volume = {77}, editor = {Aronov, Boris and Katz, Matthew J.}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.SoCG.2017.40}, URN = {urn:nbn:de:0030-drops-72289}, doi = {10.4230/LIPIcs.SoCG.2017.40}, annote = {Keywords: VC-dimension, approximation algorithms, fractional covering, multiplicative weights update, art gallery problem, polyhedral separators, geometric cove} }
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