Packing and covering with non-piercing regions

Sathish Govindarajan, Rajiv Raman, Saurabh Ray, Aniket Basu Roy

Research output: Chapter in Book/Report/Conference proceedingConference contribution


In this paper, we design the first polynomial time approximation schemes for the Set Cover and Dominating Set problems when the underlying sets are non-piercing regions (which include pseudodisks). We show that the local search algorithm that yields PTASs when the regions are disks [5, 19, 28] can be extended to work for non-piercing regions. While such an extension is intuitive and natural, attempts to settle this question have failed even for pseudodisks. The techniques used for analysis when the regions are disks rely heavily on the underlying geometry, and do not extend to topologically defined settings such as pseudodisks. In order to prove our results, we introduce novel techniques that we believe will find applications in other problems. We then consider the Capacitated Region Packing problem. Here, the input consists of a set of points with capacities, and a set of regions. The objective is to pick a maximum cardinality subset of regions so that no point is covered by more regions than its capacity. We show that this problem admits a PTAS when the regions are k-admissible regions (pseudodisks are 2-admissible), and the capacities are bounded. Our result settles a conjecture of Har-Peled (see Conclusion of [20]) in the affirmative. The conjecture was for a weaker version of the problem, namely when the regions are pseudodisks, the capacities are uniform, and the point set consists of all points in the plane. Finally, we consider the Capacitated Point Packing problem. In this setting, the regions have capacities, and our objective is to find a maximum cardinality subset of points such that no region has more points than its capacity. We show that this problem admits a PTAS when the capacity is unity, extending one of the results of Ene et al. [16].

Original languageEnglish (US)
Title of host publication24th Annual European Symposium on Algorithms, ESA 2016
EditorsChristos Zaroliagis, Piotr Sankowski
PublisherSchloss Dagstuhl- Leibniz-Zentrum fur Informatik GmbH, Dagstuhl Publishing
ISBN (Electronic)9783959770156
StatePublished - Aug 1 2016
Event24th Annual European Symposium on Algorithms, ESA 2016 - Aarhus, Denmark
Duration: Aug 22 2016Aug 24 2016

Publication series

NameLeibniz International Proceedings in Informatics, LIPIcs
ISSN (Print)1868-8969


Other24th Annual European Symposium on Algorithms, ESA 2016


  • Approximation algorithms
  • Capacitated packing
  • Dominating set
  • Local search
  • Set cover

ASJC Scopus subject areas

  • Software


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