Settling the APX-Hardness Status for Geometric Set Cover

Nabil Mustafa (1) , Rajiv Raman (2) , Saurabh Ray (3)

1 LIGM - Laboratoire d'Informatique Gaspard-Monge
2 IIT Delhi - Indian Institute of Technology Delhi
3 NYU

—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 + 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 R 3 , 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(2 polylog(n)). Together with the recent work of Chan-Grant (CGTA 2014), this settles the APX-hardness status for all natural geometric set-cover problems.

Mots clés: Hitting Sets; Pseudodisks; k-admissible regions; Quasi PTAS

Domaines: Informatique

Nabil Mustafa, Rajiv Raman, Saurabh Ray. Settling the APX-Hardness Status for Geometric Set Cover. Proc. of the 55th Annual Symposium on Foundations of Computer Science (FOCS), 2014, Philadelphia, United States. ⟨10.1109/FOCS.2014.64⟩. ⟨hal-01188991⟩