Settling the APX-hardness status for geometric set cover

Nabil H. Mustafa, Rajiv Raman, Saurabh Ray

Research output: Chapter in Book/Report/Conference proceedingConference contribution

Abstract

Weighted geometric set-cover problems arise naturally in several geometric and non-geometric settings (e.g. The breakthrough of Bansal and Pruhs (FOCS 2010) reduces a wide class of machine scheduling problems to weighted geometric set-cover). More than two decades of research has succeeded in settling the (1+ε)-approximability status for most geometric set-cover problems, except for four basic scenarios which are still lacking. One is that of weighted disks in the plane for which, after a series of papers, Varadarajan (STOC 2010) presented a clever quasi-sampling technique, which together with improvements by Chan et al(SODA 2012), yielded a O(1)-approximation algorithm. Even for the unweighted case, a PTAS for a fundamental class of objects called pseudodisks (which includes disks, unit-height rectangles, translates of convex sets etc.) is currently unknown. Another fundamental case is weighted halfspaces in R3, for which a PTAS is currently lacking. In this paper, we present a QPTAS for all of these remaining problems. Our results are based on the separator framework of Adamaszek and Wiese (FOCS 2013, SODA 2014), who recently obtained a QPTAS for weighted independent set of polygonal regions. This rules out the possibility that these problems are APX-hard, assuming NP DTIME(2polylog(n)). Together with the recent work of Chan-Grant (CGTA 2014), this settles the APX-hardness status for all natural geometric set-cover problems.

Original languageEnglish (US)
Title of host publicationProceedings - Annual IEEE Symposium on Foundations of Computer Science, FOCS
PublisherIEEE Computer Society
Pages541-550
Number of pages10
ISBN (Electronic)9781479965175
DOIs
StatePublished - Dec 7 2014
Event55th Annual IEEE Symposium on Foundations of Computer Science, FOCS 2014 - Philadelphia, United States
Duration: Oct 18 2014Oct 21 2014

Publication series

NameProceedings - Annual IEEE Symposium on Foundations of Computer Science, FOCS
ISSN (Print)0272-5428

Other

Other55th Annual IEEE Symposium on Foundations of Computer Science, FOCS 2014
CountryUnited States
CityPhiladelphia
Period10/18/1410/21/14

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Keywords

  • Hitting Sets
  • Pseudodisks
  • Quasi PTAS
  • k-admissible regions

ASJC Scopus subject areas

  • Computer Science(all)

Cite this

Mustafa, N. H., Raman, R., & Ray, S. (2014). Settling the APX-hardness status for geometric set cover. In Proceedings - Annual IEEE Symposium on Foundations of Computer Science, FOCS (pp. 541-550). [6979039] (Proceedings - Annual IEEE Symposium on Foundations of Computer Science, FOCS). IEEE Computer Society. https://doi.org/10.1109/FOCS.2014.64