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.

翻译：假设平面上给定一组蓝点集$\cal B$和一组红点集$\cal R$，所有点均位于水平线$\ell$上方。设点$p_i\in {\cal B}\cup{\cal R}$的权重为$w_i>0$（若$p_i\in {\cal B}$）或$w_i<0$（若$p_i\in {\cal R}$），$|{\cal B}\cup{\cal R}|=n$，且$d^0$（$=d\setminus\partial d$）表示任意几何对象$d$的内部。我们希望放置$k$个互不重叠的等半径圆盘$d_1,d_2,\ldots,d_k$，其圆心位于$\ell$上，并使得$\sum\limits_{j=1}^k\sum\limits_{\{i:\exists p_i\in{\cal R}, p_i\in d_j^0\}}w_i+\sum\limits_{j=1}^k\sum\limits_{\{i:\exists p_i\in{\cal B}, p_i\in d_j\}}w_i$最大化，即被$\bigcup\limits_{j=1}^kd_j$覆盖的点的权重之和最大。这里，圆盘代表产生 nuisance或损害（量值为$w_i$）的厌恶型或非理想设施，作用于其内部的需求点（如人口中心）$p_i\in {\cal R}$；反之，它们是对覆盖的需求点$p_i\in {\cal B}$提供服务的理想设施（服务量为$w_i$）。线$\ell$代表一条笔直的高速公路或铁路线。这$k$个半厌恶型设施需建在$\ell$上，以在最大化对附近吸引型需求点总服务的同时，最小化对附近排斥型需求点的损害。我们证明该问题可在$O(n^4k^2)$时间内最优求解，随后将运行时间改进至$O(n^3k \cdot\max{(\log n, k)})$。上述带权变体（即定位$k$个半厌恶型设施问题）可推广Bereg等人(2015)研究的$k=1$情形（即在线段上寻找最小半径的最大权重圆）。此外，我们还处理了问题中点的权重并非任意取值的两种特例。