The $p$-center problem (pCP) is a fundamental problem in location science, where we are given customer demand points and possible facility locations, and we want to choose $p$ of these locations to open a facility such that the maximum distance of any customer demand point to its closest open facility is minimized. State-of-the-art solution approaches of pCP use its connection to the set cover problem to solve pCP in an iterative fashion by repeatedly solving set cover problems. The classical textbook integer programming (IP) formulation of pCP is usually dismissed due to its size and bad linear programming (LP)-relaxation bounds. We present a novel solution approach that works on a new IP formulation that can be obtained by a projection from the classical formulation. The formulation is solved by means of branch-and-cut, where cuts for demand points are iteratively generated. Moreover, the formulation can be strengthened with combinatorial information to obtain a much tighter LP-relaxation. In particular, we present a novel way to use lower bound information to obtain stronger cuts. We show that the LP-relaxation bound of our strengthened formulation has the same strength as the best known bound in literature, which is based on a semi-relaxation. Finally, we also present a computational study on instances from the literature with up to more than 700000 customers and locations. Our solution algorithm is competitive with highly sophisticated set-cover-based solution algorithms, which depend on various components and parameters.
翻译:$p美元中心问题(pCP)是定位科学的一个根本问题,我们在这里得到客户需求点和可能的设施地点,我们希望从这些地点中选择美元,以打开一个设施,使客户需求的最大距离与最接近的开放设施最小化。PCP的最先进的解决方案方法利用它与设定点的联系,反复解决设定的问题,从而反复解决覆盖问题,从而解决 PCP问题。典型的教科书整型编程(IP)的制定通常会因其规模大小和线性编程(LP)不良松绑框而被忽略。我们展示了一种新的解决方案,通过古典配方的预测可以找到新的IP配方。这种配方是通过分支和切割手段解决的,在此过程中,需求点的削减是迭接的。此外,可以用组合式信息加强配方,以更紧密的LP-LP松绑定(IP)编程(IP),特别是,我们展示了一种使用更低约束性信息获得更强的裁剪裁的新型方法。我们展示了LP-松绑定的配方配置方法,而我们的精细的配法则以我们所了解的精细的精细的版本为基础,最后的算方法也是我们文献的精细的精细的精细的精细的版本。