The discrete $\alpha$-neighbor $p$-center problem (d-$\alpha$-$p$CP) is an emerging variant of the classical $p$-center problem which recently got attention in literature. In this problem, we are given a discrete set of points and we need to locate $p$ facilities on these points in such a way that the maximum distance between each point where no facility is located and its $\alpha$-closest facility is minimized. The only existing algorithms in literature for solving the d-$\alpha$-$p$CP are approximation algorithms and two recently proposed heuristics. In this work, we present two integer programming formulations for the d-$\alpha$-$p$CP, together with lifting of inequalities, valid inequalities, inequalities that do not change the optimal objective function value and variable fixing procedures. We provide theoretical results on the strength of the formulations and convergence results for the lower bounds obtained after applying the lifting procedures or the variable fixing procedures in an iterative fashion. Based on our formulations and theoretical results, we develop branch-and-cut (B&C) algorithms, which are further enhanced with a starting heuristic and a primal heuristic. We evaluate the effectiveness of our B&C algorithms using instances from literature. Our algorithms are able to solve 116 out of 194 instances from literature to proven optimality, with a runtime of under a minute for most of them. By doing so, we also provide improved solution values for 116 instances.

翻译：离散$\alpha$-邻域$p$-中心问题（d-$\alpha$-$p$CP）是经典$p$-中心问题的一个新兴变体，近期在文献中受到关注。在该问题中，给定一个离散点集，我们需要在这些点上选址$p$个设施，使得每个未设置设施的点与其第$\alpha$近设施之间的最大距离最小化。目前文献中仅有的求解d-$\alpha$-$p$CP的算法是近似算法和两种近期提出的启发式方法。本文提出了两种d-$\alpha$-$p$CP的整数规划模型，并引入了不等式提升、有效不等式、不改变最优目标函数值的不等式以及变量固定过程。我们给出了关于模型强度的理论结果，以及迭代应用提升过程或变量固定过程后所得下界的收敛性结论。基于我们的模型和理论结果，我们开发了分支切割（B&C）算法，并通过初始启发式和原始启发式进一步增强其性能。利用文献中的实例，我们评估了B&C算法的有效性。我们的算法能够求解文献中194个实例中的116个至最优性，其中大多数实例的运行时间不超过一分钟。同时，我们为这116个实例提供了改进的解值。