Line-Constrained $k$-Semi-Obnoxious Facility Location

TL;DR

This paper presents an optimal algorithm for placing multiple semi-obnoxious facilities on a line to maximize net benefit, considering both positive and negative point weights, with improved computational complexity.

Contribution

It introduces an optimal solution for the multi-facility placement problem with weighted points and improves the algorithm's efficiency from $O(n^4k^2)$ to $O(n^3k imes ext{max}( ext{log} n, k))$.

Findings

Optimal placement algorithm with $O(n^4k^2)$ complexity.

Improved algorithm with $O(n^3k imes ext{max}( ext{log} n, k))$ complexity.

Generalizes previous single-facility problem to multiple facilities.

Abstract

Suppose we are given a set $\cal B$ of blue points and a set $\cal R$ of red points, all lying above a horizontal line $\ell$, in the plane. Let the weight of a given point $p_i\in {\cal B}\cup{\cal R}$ be $w_i>0$ if $p_i\in {\cal B}$ and $w_i<0$ if $p_i\in {\cal R}$, $|{\cal B}\cup{\cal R}|=n$, and $d^0$($=d\setminus\partial d$) be the interior of any geometric object $d$. We wish to pack $k$ non-overlapping congruent disks $d_1$, $d_2$, \ldots, $d_k$ of minimum radius, centered on $\ell$ such that $\sum\limits_{j=1}^k\sum\limits_{\{i:\exists p_i\in{\cal R}, p_i\in d_j^0\}}w_i+\sum\limits_{j=1}^k\sum\limits_{\{i:\exists p_i\in{\cal B}, p_i\in d_j\}}w_i$ is maximized, i.e., the sum of the weights of the points covered by $\bigcup\limits_{j=1}^kd_j$ is maximized. Here, the disks are the obnoxious or undesirable facilities generating nuisance or damage (with quantity equal to $w_i$) to every demand point (e.g., population center) $p_i\in {\cal R}$ lying in their interior. In contrast, they are the desirable facilities giving service (equal to $w_i$) to every demand point $p_i\in {\cal B}$ covered by them. The line $\ell$ represents a straight highway or railway line. These $k$ semi-obnoxious facilities need to be established on $\ell$ to receive the largest possible overall service for the nearby attractive demand points while causing minimum damage to the nearby repelling demand points. We show that the problem can be solved optimally in $O(n^4k^2)$ time. Subsequently, we improve the running time to $O(n^3k \cdot\max{(\log n, k)})$. The above-weighted variation of locating $k$ semi-obnoxious facilities may generalize the problem that Bereg et al. (2015) studied where $k=1$ i.e., the smallest radius maximum weight circle is to be centered on a line. Furthermore, we addressed two special cases of the problem where points do not have arbitrary weights.