A Survey of Approximation Algorithms for the Universal Facility Location Problem

The facility location problem is a classical combinatorial optimization problem with extensive applications spanning communication technology, economic management, traffic governance, and public services. The facility location problem is to assign a set of clients to a set of facilities such that each client connects to a facility and the total cost (open cost and connection cost) is as low as possible. Among its various models, the uncapacitated facility location (UFL) problem is the most fundamental and widely studied. However, in real-world scenarios, resource constraints often make the UFL problem insufficient, necessitating more generalized models. This investigation primarily focuses on the universal facility location (Uni-FL) problem, a generalized framework encompassing both capacitated facility location problems (with hard and soft capacity constraints) and the UFL problem. Through a systematic analysis, we examine the Uni-FL problem alongside its specialized variants: the hard capacitated facility location (HCFL) problem and soft capacitated facility location (SCFL) problem. A comprehensive survey is conducted of existing approximation algorithms and theoretical results. The relevant results of their important variants are also discussed. In addition, we propose some open questions and future research directions for this problem based on existing research.