离散$α$-邻域$p$-中心问题的精确求解方法
Exact solution approaches for the discrete $α$-neighbor $p$-center problem
AI总结：
针对离散$α$-邻域$p$-中心问题，本文提出两种整数规划公式及多种强化技术，并开发分支切割算法，在194个文献实例中成功求解116个到已证明最优性。
摘要：
离散$α$-邻域$p$-中心问题（d-$α$-$p$CP）是经典$p$-中心问题的一个新兴变体，近年来在文献中受到关注。在该问题中，给定一个离散点集，需要在这些点上选址$p$个设施，使得每个未设置设施的点到其$α$最近设施的最大距离最小化。文献中现有的求解d-$α$-$p$CP的算法仅有近似算法和两种最近提出的启发式方法。\n\n在本工作中，我们为d-$α$-$p$CP提出了两种整数规划公式，同时给出了不等式提升、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。我们提供了关于公式强度的理论结果，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果。基于我们的公式和理论结果，我们开发了分支切割（B&C）算法，并进一步通过一个初始启发式和一个原始启发式进行增强。\n\n我们使用文献中的实例评估了所提出的B&C算法的有效性。我们的算法能够将文献中194个实例中的116个求解到已证明的最优性，其中大多数实例的运行时间不到一分钟。通过这样做，我们还为116个实例提供了改进的解值。
英文摘要：
The discrete $α$-neighbor $p$-center problem (d-$α$-$p$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 $α$-closest facility is minimized. The only existing algorithms in literature for solving the d-$α$-$p$CP are approximation algorithms and two recently proposed heuristics. In this work, we present two integer programming formulations for the d-$α$-$p$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.
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