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Improved Approximation Algorithm for Set Multicover with Non-Piercing Regions

Authors Rajiv Raman, Saurabh Ray

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Author Details

Rajiv Raman
  • IIIT Delhi, India
Saurabh Ray
  • NYU Abu Dhabi, United Arab Emirates

Acknowledgements

The authors are grateful to Nabil H. Mustafa for ideas and comments.

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Rajiv Raman and Saurabh Ray. Improved Approximation Algorithm for Set Multicover with Non-Piercing Regions. In 28th Annual European Symposium on Algorithms (ESA 2020). Leibniz International Proceedings in Informatics (LIPIcs), Volume 173, pp. 78:1-78:16, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2020) https://doi.org/10.4230/LIPIcs.ESA.2020.78

Abstract

In the Set Multicover problem, we are given a set system (X,𝒮), where X is a finite ground set, and 𝒮 is a collection of subsets of X. Each element x ∈ X has a non-negative demand d(x). The goal is to pick a smallest cardinality sub-collection 𝒮' of 𝒮 such that each point is covered by at least d(x) sets from 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+ε)-approximation algorithm for the set multicover problem (P, ℛ), where P is a set of points with demands, and ℛ is a set of non-piercing regions, as well as for the set multicover problem (𝒟, P), where 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.

Subject Classification

ACM Subject Classification
  • Theory of computation → Packing and covering problems
  • Theory of computation → Computational geometry
Keywords
  • Approximation algorithms
  • geometry
  • Covering

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