AI 中文总结
研究加权连通p-中位数问题，引入基于设施节点最小生成树定义连接权重的方法，考虑设施部署成本。证明问题NP难，提出三种MILP公式，因扩展性差开发四阶段启发式方法，通过计算研究评估性能，分析多种因素对解质量的影响。
AI 中文摘要
连通p-中位数问题是经典p-中位数问题的变体，设施节点构成连通子图。本文引入加权版本，目标函数中设施连接权重由设施节点的最小生成树定义。该方法受分布式传感器网络中汇聚节点选择的启发。考虑设施固定部署成本，目标是最小化部署成本、访问成本和连接成本之和。证明该问题是NP难的，提出三种混合整数线性规划（MILP）公式，因其扩展性差，开发了基于线性规划舍入的四阶段启发式方法。通过大量计算研究评估MILP公式和22种启发式变体在不同参数设置下的性能。结果表明MILP模型在小实例上有效，但在两小时内难以解决中大型实例，而一些启发式变体能在几分钟内产生高质量解。最后分析了网络结构、大小、密度和参数p对解质量的影响，为网络设计提供了进一步见解。
英文摘要
The connected p-median problem is defined as a variant of the classical p-median problem when the facility nodes induce a connected subgraph. In this paper, we introduce the weighted version of the above problem when the weight of the facility connection in the objective function is defined by the minimum weight spanning tree of the facility nodes. This approach is motivated by the sink node selection in distributed sensor networks, in which the collected information is shared among the sink nodes through the minimum spanning tree. The weights of the graph determining the network topology of the candidate sink nodes as connection costs are distinguished from the standard access costs of the p-median problem. The fixed deployment costs for the setup of facilities are also considered. The objective is to minimize the overall cost as the sum of deployment cost, access cost and connection cost. We show that the problem is NP-hard and propose three mixed-integer linear programming (MILP) formulations adapted from the traveling salesperson problem literature. Since these formulations are poorly scalable with respect to network size, we develop a four-phase matheuristic method based on linear programming rounding. We conduct an extensive computational study to evaluate the performance of the MILP formulations and 22 variants of the matheuristic under different parameter settings. The results indicate that the MILP models perform effectively on small instances but struggle to solve medium- and large-scale instances within a two-hour time limit. In contrast, several matheuristic variants consistently produce high-quality solutions within minutes. Finally, we analyze the impact of network structure, size, density, and the parameter $p$ on solution quality, providing further insights for network design.
Comments38 pages, 5 figures, 10 tables
