Servicing Matched Client Pairs with Facilities

TL;DR

This work proposes a linear programming (LP) relaxation for this problem, and presents a refined algorithm achieving an approximation ratio of 2.218, and provides a refined algorithm achieving an approximation ratio of 2.218 for a special case where all clients are matched.

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.

Figures (1)

• Figure 1: Illustration of rerouting along an alternating path $P$. Facilities and clients are depicted by squares and circles, respectively. The alternating path is depicted by solid edges, where the blue single-line edges correspond to $P \cap M'$ and the red double-line edges correspond to $P \cap M$. The blue dotted single-line arrow represents a positive fractional assignment of each edge in $P \cap M'$ while the red dotted double-line arrow represents the rerouted assignment for each edge in $P \cap M$.

Theorems & Definitions (39)

• Lemma 2.1
• Theorem 2.2: guha1999greedy
• Proposition 2.3: byrkaaardal2010
• Proposition 2.4
• Proposition 2.5
• Proposition 2.6
• Proposition 2.7
• Lemma 4.1
• Lemma 4.2
• proof