De, M and Lahiri, A (2023) Geometric dominatingset and setcover via localsearch. In: Computational Geometry: Theory and Applications, 113 .

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Abstract
In this paper, we study two classic optimization problems: minimum geometric dominating set and set cover. In the dominatingset problem, for a given set of objects in the plane as input, the objective is to choose a minimum number of input objects such that every input object is dominated by the chosen set of objects. Here, we say that one object is dominated by another if their intersection is nonempty. For the second problem, for a given set of points and objects in the plane, the objective is to choose a minimum number of objects to cover all the points. This is a particular version of the setcover problem. Both problems have been wellstudied, subject to various restrictions on the input objects. These problems are APXhard for object sets consisting of axisparallel rectangles, ellipses, Î±fat objects of constant description complexity, and convex polygons. On the other hand, PTASs (polynomial time approximation schemes) are known for object sets consisting of disks or unit squares. Surprisingly, a PTAS was unknown even for arbitrary squares. For both problems obtaining a PTAS remains open for a large class of objects. For the dominatingset problem, we prove that a popular localsearch algorithm leads to a (1+Îµ) approximation for a family of homothets of a convex object (which includes arbitrary squares, kregular polygons, translated and scaled copies of a convex set, etc.) in nO(1/Îµ2) time. On the other hand, the same approach leads to a PTAS for the geometric covering problem when the objects are convex pseudodisks (which include disks, unit height rectangles, homothetic convex objects, etc.). Consequently, we obtain an easytoimplement approximation algorithm for both problems for a large class of objects, significantly improving the bestknown approximation guarantees. Â© 2023 Elsevier B.V.
Item Type:  Journal Article 

Publication:  Computational Geometry: Theory and Applications 
Publisher:  Elsevier B.V. 
Additional Information:  The copyright for this article belongs to the Authors. 
Keywords:  Geometry; Local search (optimization); Polynomial approximation; Set theory, Convex objects; Dominating set problems; Dominating sets; Geometric dominating set; Geometric set cover; Geometric sets; Local search; Object sets; Polynomial time approximation schemes; Setcover, Approximation algorithms 
Department/Centre:  Division of Electrical Sciences > Computer Science & Automation 
Date Deposited:  02 May 2023 09:26 
Last Modified:  02 May 2023 09:26 
URI:  https://eprints.iisc.ac.in/id/eprint/81237 
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