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Packing and Covering with Non-Piercing Regions

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Abstract

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). Earlier, PTASs were known only in the setting where the regions were disks. These techniques relied heavily on the circularity of the disks. We develop new techniques to show that a simple local search algorithm yields a PTAS for the problems on non-piercing regions. 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 by some constant. Our result settles a conjecture of Har-Peled from 2014 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. This extends a result of Ene et al. from 2012.

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Notes

  1. A polynomial time \((1+\varepsilon )\)-approximation algorithm for any \(\varepsilon > 0\).

  2. A set of simply connected regions is said to be non-piercing if for any pair A, B of regions, the sets \(A{\setminus } B\) and \(B{\setminus } A\) are connected.

  3. A QPTAS is a \((1+\varepsilon )\)-approximation algorithm whose running time is \(O(n^{\mathrm{{polylog}}(n)})\), where n is the input size.

  4. k-Admissible regions are non-piercing regions whose boundaries intersect at most k times.

  5. An embedding of a planar graph in the plane such that the vertices are points and edges are continuous curves between the end-points that are disjoint in their interior. There could be more than one arc joining the same end-points.

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Authors and Affiliations

  1. Department of Computer Science and Automation, Indian Institute of Science, Bangalore, 560 012, India

    Aniket Basu Roy & Sathish Govindarajan

  2. Indraprastha Institute of Information Technology, 110020, New Delhi, India

    Rajiv Raman

  3. New York University Abu Dhabi, PO Box 129188, Abu Dhabi, United Arab Emirates

    Rajiv Raman & Saurabh Ray

Authors
  1. Aniket Basu Roy
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  2. Sathish Govindarajan
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  3. Rajiv Raman
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  4. Saurabh Ray
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Correspondence to Aniket Basu Roy.

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Editor in Charge János Pach

Part of the work was done when the A. Basu Roy and R. Raman were visiting New York University Abu Dhabi, UAE. A preliminary version of this paper appeared in European Symposium on Algorithms, 47:1–47:17, 2016.

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Basu Roy, A., Govindarajan, S., Raman, R. et al. Packing and Covering with Non-Piercing Regions. Discrete Comput Geom 60, 471–492 (2018). https://doi.org/10.1007/s00454-018-9983-2

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  • DOI: https://doi.org/10.1007/s00454-018-9983-2

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