Servicing Matched Client Pairs with Facilities

2026-02-23

AI Summary

This study addresses the joint optimization of facility location and customer pairing, a novel composite problem that integrates uncapacitated facility location with minimum-cost maximum matching, motivated by applications in gaming and social matchmaking. The objective is to minimize total cost while simultaneously deciding facility openings and optimal client pairings. The authors present the first constant-factor approximation algorithm, leveraging a linear programming relaxation within a bifactor approximation framework. A tailored rerouting subroutine efficiently converts fractional solutions into integral ones with only a small additive cost. The algorithm achieves an approximation ratio of 3.868 in the general setting and improves to 2.218 under full-matching constraints. Moreover, the authors establish matching upper bounds on the integrality gap of the LP relaxation, confirming the tightness of their analysis.

Technology Category

• Constraint Satisfaction and Optimization: Distributed CSP/Optimization
• Search and Optimization: Combinatorial Optimization
• Planning, Routing, and Scheduling: Optimization of Spatio-temporal Systems

Application Category

• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs
• Economics, Online Markets and Human Computation: Incentives in network design for Web infrastructures and ecosystems
• Social Networks and Social Media: Social mining and social search on the Web

Abstract

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.

Problem

• Facility Location
• Matching
• Combinatorial Optimization
• Approximation Algorithm
• LP Relaxation

Innovation

• Facility Location with Matching
• LP relaxation
• approximation algorithm
• rerouting subroutine
• integrality gap