Discretization of Planar Geometric Cover Problems

TL;DR

This paper addresses discretizing the planar geometric cover problem by converting it into a finite set cover problem, providing polynomial algorithms for various shapes including polygons and disks.

Contribution

It introduces a method to reduce the continuous solution space to a finite set of canonical translates, enabling polynomial algorithms for solving the cover problem.

Findings

Finite solution space for geometric cover problem

Polynomial algorithms for disks and polygons

Applicable to convex and non-convex shapes

Abstract

We consider discretization of the 'geometric cover problem' in the plane: Given a set $P$ of $n$ points in the plane and a compact planar object $T_0$, find a minimum cardinality collection of planar translates of $T_0$ such that the union of the translates in the collection contains all the points in $P$. We show that the geometric cover problem can be converted to a form of the geometric set cover, which has a given finite-size collection of translates rather than the infinite continuous solution space of the former. We propose a reduced finite solution space that consists of distinct canonical translates and present polynomial algorithms to find the reduce solution space for disks, convex/non-convex polygons (including holes), and planar objects consisting of finite Jordan curves.