Efficient Algorithms for Obnoxious Facility Location on a Line Segment or Circle

TL;DR

This paper presents improved efficient algorithms for various restricted obnoxious facility location problems on line segments and circles, significantly reducing computational complexity compared to previous methods.

Contribution

It introduces faster algorithms for constrained obnoxious facility location on line segments and circles, and for the min-sum obnoxious facility location problem, surpassing prior results.

Findings

Algorithms run in $O(n^2 \, \log k + n \log k \log (n^2 + k))$ time for line and circle problems.

Achieves improved time complexity over previous $O((nk)^2 \log(nk) + (n + k) \log (nk))$ results.

Provides an $O(nk \alpha(nk) \log^3 {nk})$ time algorithm for min-sum obnoxious facility location.

Abstract

We study different restricted variations of the obnoxious facility location problem on a plane. The first is the constrained obnoxious facility location on a line segment (COFL-Line) problem. We provide an efficient algorithm for this problem that executes in $O(n ^ 2 \log k + n \log k \log (n^2 + k))$ time. Our result improves on the best known result of $O((nk)^2 \log(nk) + (n + k) \log (nk))$ time obtained by Singireddy and Basappa\cite{singireddy2022dispersing}. We also study the same problem where the facilities must be placed on a given circle (the constrained obnoxious facility location on a circle (COFL-Circ) problem). We provide an efficient algorithm for this problem that executes in $O(n ^ 2 \log k + n \log k \log (n^2 + k))$ time. Our result improves on the best known result of $O((nk)^2 \log(nk) + (n + k) \log (nk))$ time obtained by Singireddy and Basappa\cite{singireddy2022dispersing}. The third problem we study is the min-sum obnoxious facility location (MOFL) problem.We provide an efficient algorithm that executes in $O(nk\cdot \alpha(nk) \log^3 {nk})$ time, where $\alpha(.)$ is the inverse Ackermann function. The best known previous result is an $O(n^3k)$ time obtained by Singireddy and Basappa\cite{singireddy2022dispersing}.