$p$-中心距离与逆中心距离问题的近似性结果
Approximability results for the $p$-centdian and the converse centdian problems
AI总结：
本文证明了 $p$-中心距离问题是 NP-完全的，并首次设计了两个问题的非平凡暴力精确算法及两个近似算法，兼顾最小总距离与最大距离的服务质量。
摘要：
给定一个无向图 $G=(V,E)$，其边具有非负长度函数，以及一个整数 $p$，满足 $0 < p < |V|$，$p$-中心距离问题（$p$-centdian problem）旨在找到 $V$ 中的 $p$ 个顶点（称为中心距离集），使得离心率（eccentricity）与中位距离（median-distance）之和最小化，其中离心率是所有顶点到其最近的中心距离集的最大（长度）距离，中位距离是所有顶点到其最近的中心距离集的总（长度）距离。离心率与中位距离之和称为中心距离（centdian-distance）。$p$-中心距离问题的目的是找到 $p$ 个开放设施（服务器），使其同时满足服务客户的最小总距离（中位距离）和最大距离（离心率）的服务质量。如果我们将这两个准则互换，即给定中心距离的界限，目标函数是最小化中心距离集的基数，则此问题称为逆中心距离问题（converse centdian problem）。在本文中，我们证明了 $p$-中心距离问题是 NP-完全的。然后，我们分别为 $p$-中心距离问题和逆中心距离问题设计了第一个非平凡的暴力精确算法。最后，我们为这两个问题设计了两个近似算法。
英文摘要：
Given an undirected graph $G=(V,E)$ with a nonnegative edge length function and an integer $p$, $0 < p < |V|$, the $p$-centdian problem is to find $p$ vertices (called the {\it centdian set}) of $V$ such that the {\it eccentricity} plus {\it median-distance} is minimized, in which the {\it eccentricity} is the maximum (length) distance of all vertices to their nearest {\it centdian set} and the {\it median-distance} is the total (length) distance of all vertices to their nearest {\it centdian set}. The {\it eccentricity} plus {\it median-distance} is called the {\it centdian-distance}. The purpose of the $p$-centdian problem is to find $p$ open facilities (servers) which satisfy the quality-of-service of the minimum total distance ({\it median-distance}) and the maximum distance ({\it eccentricity}) to their service customers, simultaneously. If we converse the two criteria, that is given the bound of the {\it centdian-distance} and the objective function is to minimize the cardinality of the {\it centdian set}, this problem is called the converse centdian problem. In this paper, we prove the $p$-centdian problem is NP-Complete. Then we design the first non-trivial brute force exact algorithms for the $p$-centdian problem and the converse centdian problem, respectively. Finally, we design two approximation algorithms for both problems.
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。
