Quasi-polynomial time approxim ation scheme for weighted geometric set cover on pseudodisks and halfspaces

Nabil H. Mustafa, Rajiv Raman, Saurabh Ray

Research output: Contribution to journalArticle

Abstract

Weighted geometric set-cover problems arise naturally in several geometric and nongeometric settings (e.g., the breakthrough of Bansal and Pruhs [Proceedings of FOCS, 2010, pp. 407-414] 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 + ∈)-approximability status for most geometric set-cover problems, except for some basic scenarios which are still lacking. One is that of weighted disks in the plane for which, after a series of papers, Varadarajan [Proceedings of STOC'10, 2010, pp. 641-648] presented a clever quasi-sampling technique, which together with improvements by Chan et al. [Proceedings of SODA, 2012, pp. 1576-1585], yielded an O(1)-approximation algorithm. Even for the unweighted case, a polynomial time approximation scheme (PTAS) for a fundamental class of objects called pseudodisks (which includes halfspaces, 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 quasi PTAS (QPTAS) for all these remaining problems. Our results are based on the separator framework of Adamaszek and Wiese [Proceedings of FOCS, 2013, pp. 400-409; Proceedings of SODA, 2014, pp. 645-656], who recently obtained a QPTAS for a 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 and Grant [Comput. Geom., 47(2014), pp. 112-124], this settles the APX-hardness status for all natural geometric set-cover problems.

Original languageEnglish (US)
Pages (from-to)1650-1669
Number of pages20
JournalSIAM Journal on Computing
Volume44
Issue number6
DOIs
StatePublished - Jan 1 2015

Fingerprint

Set Cover
Half-space
Polynomial time
Polynomial Time Approximation Scheme
Polynomials
Approximation algorithms
Separators
Hardness
Scheduling
Machine Scheduling
Approximability
Separator
Sampling
Independent Set
Rectangle
Convex Sets
Unit Disk
Scheduling Problem
Approximation Algorithms
NP-complete problem

Keywords

  • Approximation algorithms
  • Polynomial time approximation schemes
  • Pseudodisks
  • Set cover

ASJC Scopus subject areas

  • Computer Science(all)
  • Mathematics(all)

Cite this

Quasi-polynomial time approxim ation scheme for weighted geometric set cover on pseudodisks and halfspaces. / Mustafa, Nabil H.; Raman, Rajiv; Ray, Saurabh.

In: SIAM Journal on Computing, Vol. 44, No. 6, 01.01.2015, p. 1650-1669.

Research output: Contribution to journalArticle

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