Polynomial-time approximation schemes for piercing and covering with applications in wireless networks

Paz Carmi, Matthew J. Katz, Nissan Lev-Tov

Research output: Contribution to journalArticlepeer-review

8 Scopus citations

Abstract

Let D be a set of disks of arbitrary radii in the plane, and let P be a set of points. We study the following three problems: (i) Assuming P contains the set of center points of disks in D, find a minimum-cardinality subset P of P (if exists), such that each disk in D is pierced by at least h points of P, where h is a given constant. We call this problem minimum h-piercing. (ii) Assuming P is such that for each D∈D there exists a point in P whose distance from D's center is at most αr(D), where r(D) is D's radius and 0≤α<1 is a given constant, find a minimum-cardinality subset P of P, such that each disk in D is pierced by at least one point of P. We call this problem minimum discrete piercing with cores. (iii) Assuming P is the set of center points of disks in D, and that each D∈D covers at most l points of P, where l is a constant, find a minimum-cardinality subset D of D, such that each point of P is covered by at least one disk of D. We call this problem minimum center covering. For each of these problems we present a constant-factor approximation algorithm (trivial for problem (iii)), followed by a polynomial-time approximation scheme. The polynomial-time approximation schemes are based on an adapted and extended version of Chan's [T.M. Chan, Polynomial-time approximation schemes for packing and piercing fat objects, J. Algorithms 46 (2003) 178-189] separator theorem. Our PTAS for problem (ii) enables one, in practical cases, to obtain a (1+)-approximation for minimum discrete piercing (i.e., for arbitrary P).

Original languageEnglish
Pages (from-to)209-218
Number of pages10
JournalComputational Geometry: Theory and Applications
Volume39
Issue number3
DOIs
StatePublished - 1 Apr 2008

Keywords

  • Approximation algorithms
  • Covering
  • Discrete piercing
  • Geometric optimization
  • Wireless networks

ASJC Scopus subject areas

  • Computer Science Applications
  • Geometry and Topology
  • Control and Optimization
  • Computational Theory and Mathematics
  • Computational Mathematics

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