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On the Geometric Set Multicover Problem

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Abstract

In the Set Multicover problem, we are given a set system \((X,{\mathcal {S}})\), where X is a finite ground set, and \({\mathcal {S}}\) is a collection of subsets of X. Each element \(x \in X\) has a non-negative demand d(x). The goal is to pick a smallest cardinality sub-collection \({\mathcal {S}}'\) of \({\mathcal {S}}\) such that each point is covered by at least d(x) sets from \({\mathcal {S}}'\). In this paper, we study the set multicover problem for set systems defined by points and non-piercing regions in the plane, which includes disks, pseudodisks, k-admissible regions, squares, unit height rectangles, homothets of convex sets, upward paths on a tree, etc. We give a polynomial time \((2+\epsilon )\)-approximation algorithm for the set multicover problem \((P,{\mathcal {R}})\), where P is a set of points with demands, and \({\mathcal {R}}\) is a set of non-piercing regions, as well as for the set multicover problem \(({\mathcal {D}}, P)\), where \({\mathcal {D}}\) is a set of pseudodisks with demands, and P is a set of points in the plane, which is the hitting set problem with demands.

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Notes

  1. In the weighted setting, the sets have non-negative weights, and the goal is to find a minimum weight feasible sub-collection, as opposed to the minimum cardinality.

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Authors and Affiliations

  1. IIIT-Delhi, Delhi, India

    Rajiv Raman

  2. New York University, Abu Dhabi, UAE

    Saurabh Ray

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  1. Rajiv Raman
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  2. Saurabh Ray
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Correspondence to Rajiv Raman.

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Raman, R., Ray, S. On the Geometric Set Multicover Problem. Discrete Comput Geom 68, 566–591 (2022). https://doi.org/10.1007/s00454-022-00402-y

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