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：假设在平面上给定一组位于水平直线 $\ell$ 上方的蓝点集 $\cal B$ 和红点集 $\cal R$。设点 $p_i\in {\cal B}\cup{\cal R}$ 的权重为 $w_i$，其中若 $p_i\in {\cal B}$ 则 $w_i>0$，若 $p_i\in {\cal R}$ 则 $w_i<0$，且 $|{\cal B}\cup{\cal R}|=n$，同时用 $d^0$($=d\setminus\partial d$) 表示任意几何对象 $d$ 的内部。我们希望将 $k$ 个互不重叠且半径最小的等大圆盘 $d_1, d_2, \ldots, d_k$ 中心置于直线 $\ell$ 上，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 $\bigcup\limits_{j=1}^kd_j$ 所覆盖点的权重之和最大化。在此，圆盘被视为产生滋扰或损害（数量等于 $w_i$）的厌恶型或非理想设施，针对位于其内部的所有需求点（如人口中心）$p_i\in {\cal R}$；反之，它们则为提供服务的理想设施（服务量等于 $w_i$），面向所覆盖的需求点 $p_i\in {\cal B}$。直线 $\ell$ 代表一条笔直的公路或铁路线。这 $k$ 个半厌恶型设施需设立于 $\ell$ 上，以便为邻近的吸引型需求点提供最大整体服务，同时对邻近的排斥型需求点造成最小损害。我们证明该问题可在 $O(n^4k^2)$ 时间内最优求解。随后，我们将运行时间改进至 $O(n^3k \cdot\max{(\log n, k)})$。上述带权重的 $k$ 个半厌恶型设施选址变体问题可推广 Bereg 等人 (2015) 研究的 $k=1$ 情形，即寻找圆心位于直线上且半径最小的最大权重圆。此外，我们还探讨了该问题的两种特殊情况，其中点不具有任意权重。