Servicing Matched Client Pairs with Facilities

TL;DR

This paper introduces a new Facility Location with Matching problem, develops LP-based approximation algorithms with ratios around 3.87 and 2.22 for special cases, and extends existing optimization frameworks.

Contribution

It proposes the first LP relaxation and approximation algorithms for Facility Location with Matching, generalizing previous problems and providing bounds on integrality gaps.

Findings

Achieves a 3.868-approximation algorithm for the general problem.

Provides a 2.218-approximation for the all-matched clients case.

Establishes bounds on the LP relaxation's integrality gap.

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.