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International Workshop on Combinatorial Algorithms

IWOCA 2021: Combinatorial Algorithms pp 295–309Cite as

Piercing All Translates of a Set of Axis-Parallel Rectangles

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Part of the book series: Lecture Notes in Computer Science ((LNTCS,volume 12757))

Abstract

For a given shape S in the plane, one can ask what is the lowest possible density of a point set P that pierces (“intersects”, “hits”) all translates of S. This is equivalent to determining the covering density of S and as such is well studied. Here we study the analogous question for families of shapes where the connection to covering no longer exists. That is, we require that a single point set P simultaneously pierces each translate of each shape from some family \(\mathcal F\). We denote the lowest possible density of such an \(\mathcal F\)-piercing point set by \(\pi _T(\mathcal F)\). Specifically, we focus on families \(\mathcal F\) consisting of axis-parallel rectangles. When \(|\mathcal F|=2\) we exactly solve the case when one rectangle is more squarish than \(2\times 1\), and give bounds (within \(10\%\) of each other) for the remaining case when one rectangle is wide and the other one is tall. When \(|\mathcal F|\ge 2\) we present a linear-time constant-factor approximation algorithm for computing \(\pi _T(\mathcal F)\) (with ratio 1.895).

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Acknowledgments

We would like to thank Wolfgang Mulzer and Jakub Svoboda for helpful comments on an earlier version of this work.

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Authors and Affiliations

  1. Algoresearch L.L.C., Milwaukee, USA

    Adrian Dumitrescu

  2. Department of Mathematics, Harvard University, Cambridge, MA, 02138, USA

    Josef Tkadlec

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  1. Adrian Dumitrescu
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Editors and Affiliations

  1. University of Ottawa, Ottawa, ON, Canada

    Paola Flocchini

  2. University of Ottawa, Ottawa, ON, Canada

    Lucia Moura

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Dumitrescu, A., Tkadlec, J. (2021). Piercing All Translates of a Set of Axis-Parallel Rectangles. In: Flocchini, P., Moura, L. (eds) Combinatorial Algorithms. IWOCA 2021. Lecture Notes in Computer Science(), vol 12757. Springer, Cham. https://doi.org/10.1007/978-3-030-79987-8_21

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  • DOI: https://doi.org/10.1007/978-3-030-79987-8_21

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  • Online ISBN: 978-3-030-79987-8

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