We introduce the conditional p-dispersion problem (c-pDP), an incremental variant of the p-dispersion problem (pDP). In the c-pDP, one is given a set N of n points, a symmetric dissimilarity matrix D of dimensions \(n\times n\), an integer \(p\ge 1\) and a set \(Q\subseteq N\) of cardinality \(q\ge 1\). The objective is to select a set \(P\subset N\setminus Q\) of cardinality p that maximizes the minimal dissimilarity between every pair of selected vertices, i.e., \(z(P\cup Q) {:}{=}\min \{D(i, j), i, j\in P\cup Q\}\). The set Q may model a predefined subset of preferences or hard location constraints in incremental network design. We adapt the state-of-the-art algorithm for the pDP to the c-pDP and include an ad-hoc acceleration mechanism designed to leverage the information provided by the set Q to further reduce the size of the problem instance. We perform exhaustive computational experiments and show that the proposed acceleration mechanism helps reduce the total computational time by a factor of five on average. We also assess the scalability of the algorithm and derive sensitivity analyses.

我们引入有条件的p -分散的问题（C-PDP），该的增量变种p -分散的问题（PDP）。在c-pDP中，给定N个点的集合N，维\（n \ timesn \）的对称不相似矩阵D，整数\（p \ ge 1 \）和集合\（Q \ subseteq N \）基数\（q \ ge 1 \）。目的是选择基数p的集合\（P \ subset N \ setminus Q \），该集合最大程度地增加每对选定顶点之间的最小相似度，即\（z（P \ cup Q）{：} {=} \ min \ {D（i，j），i，j \ in P \ cup Q \} \）。集合Q可以在增量网络设计中对偏好或硬位置约束的预定义子集建模。我们将pDP的最新算法与c-pDP相适应，并包括一个临时加速机制，该机制旨在利用集合Q提供的信息来进一步减小问题实例的大小。我们进行了详尽的计算实验，并证明了所提出的加速机制平均可将总计算时间减少五倍。我们还评估了该算法的可扩展性，并得出了灵敏度分析。