Dispersing Facilities on Planar Segment and Circle Amidst Repulsion

TL;DR

This paper develops efficient algorithms for optimally placing non-overlapping disks amidst demand points on a line segment or circle boundary, improving computational speed and providing exact solutions for specific geometric constraints.

Contribution

It introduces exact polynomial-time algorithms for the problem constrained to a line segment, improves existing methods with faster parametric search, and adapts approximation algorithms for circular constraints.

Findings

Exact polynomial-time algorithm for line segment constrained problem.

Faster $O((n+k)^2)$-time algorithm using parametric search.

Efficient adaptation of approximation algorithm for circular boundary constraints.

Abstract

In this paper we consider the problem of locating $k$ obnoxious facilities (congruent disks of maximum radius) amidst $n$ demand points (existing repulsive facility sites) ordered from left to right in the plane so that none of the existing facility sites are affected (no demand point falls in the interior of the disks). We study this problem in two restricted settings: (i) the obnoxious facilities are constrained to be centered on along a predetermined horizontal line segment $\bar{pq}$, and (ii) the obnoxious facilities are constrained to lie on the boundary arc of a predetermined disk $\cal C$. An $(1-\epsilon)$-approximation algorithm was given recently to solve the constrained problem in (i) in time $O((n+k)\log{\frac{||pq||}{2(k-1)\epsilon}})$, where $\epsilon>0$ \cite{Sing2021}. Here, for the problem in (i), we first propose an exact polynomial-time algorithm based on a binary search on all candidate radii computed explicitly. This algorithm runs in $O((nk)^2\log{(nk)}+(n+k)\log{(nk)})$ time. We then show that using the parametric search technique of Megiddo \cite{MG1983}; we can solve the problem exactly in $O((n+k)^2)$ time, which is faster than the latter. Continuing further, using the improved parametric technique we give an $O(n\log^2 n)$-time algorithm for $k=2$. We finally show that the above $(1-\epsilon)$-approximation algorithm of \cite{Sing2021} can be easily adapted to solve the circular constrained problem of (ii) with an extra multiplicative factor of $n$ in the running time.