Maximum Centre-Disjoint Mergeable Disks

TL;DR

This paper studies a geometric problem involving selecting and merging disks to avoid center containment, proving NP-hardness, providing an ILP formulation, and offering a polynomial-time solution for collinear centers.

Contribution

It introduces the maximum centre-disjoint mergeable disks problem, proves its NP-hardness, and provides both an ILP model and an efficient algorithm for a special case.

Findings

The problem is NP-hard.

An ILP formulation is provided.

A polynomial-time algorithm exists for collinear centers.

Abstract

Given a set of disks in the plane, the goal of the problem studied in this paper is to choose a subset of these disks such that none of its members contains the centre of any other. Each disk not in this subset must be merged with one of its nearby disks that is, increasing the latter's radius. This problem has applications in labelling rotating maps and in visualizing the distribution of entities in static maps. We prove that this problem is NP-hard. We also present an ILP formulation for this problem, and a polynomial-time algorithm for the special case in which the centres of all disks are on a line.