We study Facility Location with Matching, a Facility Location problem where, given additional information about which pair of clients is compatible to be matched, we need to match as many clients as possible and assign each matched client pair to a same open facility at minimum total cost. The problem is motivated by match-making services relevant in, for example, video games or social apps. It naturally generalizes two prominent combinatorial optimization problems -- Uncapacitated Facility Location and Minimum-cost Maximum Matching. Facility Location with Matching also generalizes the Even-constrained Facility Location problem studied by Kim, Shin, and An (Algorithmica 2023). We propose a linear programming (LP) relaxation for this problem, and present a 3.868-approximation algorithm. Our algorithm leverages the work on bifactor-approximation algorithms (Byrka and Aardal, SICOMP 2012); our main technical contribution is a rerouting subroutine that reroutes a fractional solution to be supported on a fixed maximum matching with only small additional cost. For a special case where all clients are matched, we provide a refined algorithm achieving an approximation ratio of 2.218. As our algorithms are based on rounding an optimal solution to the LP relaxation, these approximation results also give the same upper bounds on the integrality gap of the relaxation.

翻译：本研究探讨带匹配的设施选址问题，该问题在给定客户间兼容匹配对信息的基础上，要求最大化匹配客户对数量，并将每个匹配成功的客户对分配至同一开放设施，同时最小化总成本。该问题源于视频游戏或社交应用等场景中的匹配服务需求。它自然推广了两个重要的组合优化问题——无容量限制设施选址问题与最小成本最大匹配问题。带匹配的设施选址问题还推广了Kim、Shin与An（Algorithmica 2023）研究的偶数约束设施选址问题。我们提出了该问题的线性规划松弛模型，并给出了3.868倍近似算法。该算法基于双因子近似算法框架（Byrka与Aardal，SICOMP 2012）；我们的核心技术贡献在于提出了一种重路由子程序，该程序能将分数解重路由至固定的最大匹配支撑集上，且仅产生微小附加成本。针对所有客户均需匹配的特殊情形，我们提出了改进算法，获得2.218的近似比。由于算法基于对线性规划松弛最优解的舍入处理，这些近似结果也给出了该松弛整数间隙的相同上界。