Efficient covering of convex domains by congruent discs

TL;DR

This paper investigates the problem of covering convex regions with congruent discs, providing improved bounds, algorithms with approximation guarantees, and analyzing their computational complexity.

Contribution

It introduces new bounds and algorithms for covering convex domains with congruent discs, including approximation guarantees and complexity analysis.

Findings

Improved upper and lower bounds on disc covering numbers

Algorithms with constant factor approximation guarantees

Pseudo-polynomial complexity of the proposed algorithms

Abstract

In this paper, we consider the problem of covering a plane region with unit discs. We present an improved upper bound and the first nontrivial lower bound on the number of discs needed for such a covering, depending on the area and perimeter of the region. We provide algorithms for efficient covering of convex polygonal regions using unit discs. We show that the computational complexity of the algorithms is pseudo-polynomial in the size of the input and the output. We also show that these algorithms provide a constant factor approximation of the optimal covering of the region.