TY - JOUR

T1 - Quasi-polynomial time approxim ation scheme for weighted geometric set cover on pseudodisks and halfspaces

AU - Mustafa, Nabil H.

AU - Raman, Rajiv

AU - Ray, Saurabh

N1 - Publisher Copyright:
Copyright © by SIAM.

PY - 2015

Y1 - 2015

N2 - 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.

AB - 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.

KW - Approximation algorithms

KW - Polynomial time approximation schemes

KW - Pseudodisks

KW - Set cover

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UR - http://www.scopus.com/inward/citedby.url?scp=84971668377&partnerID=8YFLogxK

U2 - 10.1137/14099317X

DO - 10.1137/14099317X

M3 - Article

AN - SCOPUS:84971668377

SN - 0097-5397

VL - 44

SP - 1650

EP - 1669

JO - SIAM Journal on Computing

JF - SIAM Journal on Computing

IS - 6

ER -