Geometric Dominating Set and Set Cover via Local Search
Minati De, Abhiruk Lahiri

TL;DR
This paper develops local search algorithms that provide near-optimal solutions for geometric dominating set and set cover problems involving convex objects, with improved approximation guarantees and practical efficiency.
Contribution
It introduces a local search-based PTAS for geometric set cover and dominating set problems with convex objects, expanding the classes of objects for which efficient approximations are known.
Findings
Local search yields a (1+ε)-approximation for homothetic convex objects.
PTAS achieved for convex pseudodisks covering points.
Significant improvement over previous approximation guarantees.
Abstract
In this paper, we study two classic optimization problems: minimum geometric dominating set and set cover. In the dominating-set 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, one object is dominated by {another} if both of them have {a} nonempty intersection region. For the second problem, for a given set of points and objects {in a plane}, the objective is to choose {a} minimum number of objects to cover all the points. This is a special version of the set-cover problem. For both problems obtaining a PTAS remains open for a large class of objects. For the dominating-set problem, we prove that {a} popular local-search algorithm leads to an approximation for object sets consisting of homothetic set of convex…
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Taxonomy
TopicsComputational Geometry and Mesh Generation · Advanced Graph Theory Research · Complexity and Algorithms in Graphs
