SCIENCE@home

QPTAS for Geometric Set-Cover Problems via Optimal Separators

Nabil H. Mustafa,R. Raman,Saurabh Ray

Published 2014 in arXiv.org

ABSTRACT

Weighted geometric set-cover problems arise naturally in several geometric and non-geometric settings (e.g. the breakthrough of Bansal-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 $(1+\epsilon)$-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 \emph{quasi-sampling} technique, which together with improvements by Chan \etal~(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 $\Re^3$, 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-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 $\textbf{NP} \nsubseteq \textbf{DTIME}(2^{polylog(n)})$. Together with the recent work of Chan-Grant (CGTA 2014), this settles the APX-hardness status for all natural geometric set-cover problems.

PUBLICATION RECORD

  • Publication year

    2014

  • Venue

    arXiv.org

  • Publication date

    2014-03-04

  • Fields of study

    Mathematics, Computer Science

  • Identifiers
    arXiv 1403.0835
  • External record

    Open on Semantic Scholar

  • Source metadata

    Semantic Scholar

CITATION MAP

EXTRACTION MAP

CLAIMS

  • No claims are published for this paper.

CONCEPTS

  • No concepts are published for this paper.

REFERENCES

Showing 1-33 of 33 references · Page 1 of 1

CITED BY

Showing 1-17 of 17 citing papers · Page 1 of 1