直线约束下的$k$-半厌恶设施选址
Line-Constrained $k$-Semi-Obnoxious Facility Location
AI总结：
本文研究直线约束下的k半厌恶设施选址问题，提出最优求解算法并优化时间复杂度，推广了已有单设施选址研究，还探讨了点无任意权重的特殊情形。
摘要：
假设在平面中给定一组蓝点集合$\boldsymbol{\text{蓝点集合}\cal B}$和一组红点集合$\boldsymbol{\text{红点集合}\cal R}$，所有点都位于水平线$\boldsymbol{\text{直线}\ell}$的上方。对于给定点$p_i \in {\cal B}\cup{\cal R}$，若$p_i \in {\cal B}$，则其权重$w_i>0$；若$p_i \in {\cal R}$，则其权重$w_i<0$，且$|{\cal B}\cup{\cal R}|=n$，$d^0$（即$d\setminus\partial d$）表示任意几何对象$d$的内部。我们希望放置$k$个互不重叠的全等圆盘$d_1,d_2,\ldots,d_k$，这些圆盘的圆心位于$\ell$上且半径最小，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$\bigcup\limits_{j=1}^kd_j$所覆盖的点的权重之和最大。此处，这些圆盘是厌恶型或不受欢迎的设施，会对位于其内部的每个需求点（例如人口中心）$p_i \in {\cal R}$产生滋扰或损害（损害量等于$w_i$的绝对值）。相反，它们也是受欢迎的设施，会为被其覆盖的每个需求点$p_i \in {\cal B}$提供服务（服务量等于$w_i$）。直线$\boldsymbol{\text{直线}\ell}$代表一条笔直的公路或铁路线。需要在$\boldsymbol{\text{直线}\ell}$上建立这$k$个半厌恶设施，以便为附近的吸引型需求点提供尽可能多的综合服务，同时对附近的排斥型需求点造成最小的损害。我们证明该问题可以在$O(n^4k^2)$时间内最优求解。随后，我们将运行时间优化至$O(n^3k \cdot\max{(\log n, k)})$。上述加权$k$个半厌恶设施选址问题可以推广Bereg等人（2015）研究的$k=1$的情况，即圆心位于直线上的最小半径最大权重圆问题。此外，我们还讨论了该问题的两个特殊情况，即点不具有任意权重的情况。
英文摘要：
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.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
