The maximal covering location problem (MCLP) and the partial set covering location problem (PSCLP) are two fundamental problems in facility location and have widespread applications in practice. The MCLP determines a subset of facilities to open to maximize the demand of covered customers subject to a budget constraint on the cost of open facilities; and the PSCLP aims to minimize the cost of open facilities while requiring a certain amount of customer demand to be covered. Both problems can be modeled as mixed integer programming (MIP) formulations. Due to the intrinsic $\mathcal{NP}$-hardness nature, however, it is a great challenge to solve them to optimality by MIP solvers, especially for large-scale cases. In this paper, we present five customized presolving methods to enhance the capability of employing MIP solvers in solving the two problems. The five presolving methods are designed to reduce the sizes of the problem formulation and the search tree of the branch-and-cut procedure. For planar problems with an extremely huge number of customers under realistic types of facility coverage, we show that the number of customers in the reduced problems can be bounded above by a quadratic polynomial of the number of facilities. By extensive numerical experiments, the five presolving methods are demonstrated to be effective in accelerating the solution process of solving the MCLP and PSCLP. Moreover, they enable to solve problems with billions of customers, which is at least one order of magnitude larger than those that can be tackled by previous approaches.

最大覆盖位置问题（MCLP）和部分集覆盖位置问题（PSCLP）是设施选址中的两个基本问题，在实践中有着广泛的应用。MCLP 确定要开放的设施子集，以最大限度地满足所覆盖客户的需求，但须遵守开放设施成本的预算限制；PSCLP 旨在最大限度地降低开放设施的成本，同时要求满足一定量的客户需求。这两个问题都可以建模为混合整数规划 (MIP) 公式。由于内在的$\mathcal{NP}$-硬度性质，但是，通过 MIP 求解器将它们求解到最优是一个巨大的挑战，特别是对于大规模情况。在本文中，我们提出了五种定制的预求解方法，以增强使用 MIP 求解器解决这两个问题的能力。这五种预求解方法旨在减少问题表述和分支切割过程的搜索树的大小。对于现实类型的设施覆盖范围内具有大量客户的平面问题，我们表明，减少的问题中的客户数量可以由设施数量的二次多项式来限制。通过大量的数值实验，这五种预求解方法被证明可以有效地加速求解MCLP和PSCLP的求解过程。而且，