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

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}.