In this announcement, we show that the classical Maximum Coverage problem (MC) admits a PTAS via local search in essentially all cases where the corresponding instances of Set Cover (SC) admit a PTAS via the local search approach by Mustafa and Ray [Nabil H. Mustafa and Saurabh Ray, 2010]. As a corollary, we answer an open question by Badanidiyuru, Kleinberg, and Lee [Ashwinkumar Badanidiyuru et al., 2012] regarding half-spaces in R^3 thereby settling the existence of PTASs for essentially all natural cases of geometric MC problems. As an intermediate result, we show a color-balanced version of the classical planar subdivision theorem by Frederickson [Greg N. Frederickson, 1987]. We believe that some of our ideas may be useful for analyzing local search in other settings involving a hard cardinality constraint.
@InProceedings{chaplick_et_al:LIPIcs.ICALP.2018.107, author = {Chaplick, Steven and De, Minati and Ravsky, Alexander and Spoerhase, Joachim}, title = {{Brief Announcement: Approximation Schemes for Geometric Coverage Problems}}, booktitle = {45th International Colloquium on Automata, Languages, and Programming (ICALP 2018)}, pages = {107:1--107:4}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-076-7}, ISSN = {1868-8969}, year = {2018}, volume = {107}, editor = {Chatzigiannakis, Ioannis and Kaklamanis, Christos and Marx, D\'{a}niel and Sannella, Donald}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ICALP.2018.107}, URN = {urn:nbn:de:0030-drops-91113}, doi = {10.4230/LIPIcs.ICALP.2018.107}, annote = {Keywords: balanced separators, maximum coverage, local search, approximation scheme, geometric approximation algorithms} }
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