Unique Set Cover on Unit Disks and Unit Squares
Saeed Mehrabi

TL;DR
This paper investigates the computational complexity of the Unique Set Cover problem for unit disks and squares, proving NP-hardness and providing a polynomial-time approximation scheme for unit squares.
Contribution
It establishes NP-hardness for the problem on both shapes and introduces a PTAS for unit squares using the mod-one approach.
Findings
Unique Set Cover is NP-hard on unit disks and squares.
A PTAS is developed for unit squares.
The problem remains computationally challenging for these geometric objects.
Abstract
We study the Unique Set Cover problem on unit disks and unit squares. For a given set of points and a set of geometric objects both in the plane, the objective of the Unique Set Cover problem is to select a subset of objects such that every point in is covered by at least one object in and the number of points covered uniquely is maximized, where a point is covered uniquely if the point is covered by exactly one object in . In this paper, (i) we show that the Unique Set Cover is NP-hard on both unit disks and unit squares, and (ii) we give a PTAS for this problem on unit squares by applying the mod-one approach of Chan and Hu (Comput. Geom. 48(5), 2015).
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Taxonomy
TopicsComputational Geometry and Mesh Generation · Complexity and Algorithms in Graphs · Data Management and Algorithms
