Approximation algorithms for facility location problems

One of the most important aspects of logistics is to decide where to locate new facilities such as plants, distribution centers, and retailers. Facility location models not only have important applications in designing distribution systems, but also often form identifiable parts of other practical problems. However, many discrete location problems are NP-hard and the scale of the instances arising in practice is often too large to be solved optimally. In this thesis, we focus on polynomial time approximation algorithms for solving the well-known uncapacitated facility location problem and its generalizations. In the uncapacitated problem, we are given a set of clients; a set of possible locations for facilities, the cost of opening a facility at each location, and the cost of connecting each client to a facility at each location. The objective is to open facilities at a subset of these locations, and connect each client to an open facility to minimize the sum of facility opening and connection costs. We assume that connection costs obey the triangle inequality. It is known that a 1.46-approximation algorithm for the uncapacitated problem would imply P = NP. In this thesis, we improve a long line of previous results and give a 1.52-approximation algorithm for the uncapacitated problem. We also consider several important generalizations of the uncapacitated problem, including: (1) Capacitated facility location: Each facility can serve only a certain amount of clients. We present a multi-exchange local search algorithm for this problem. We show its approximation ratio is between 3 + 22 − e and 3 + 22 + e for any given constant e > 0. (2) Multi-level facility location: The demands must be routed among the facilities in a hierarchical order. We give the best combinatorial algorithm for this problem. For the special case when there are only two levels of facilities, we give a 1.77-approximation algorithm, which is currently the best. (3) Dynamic facility location: The demand varies between time-periods. In addition to the question of where to locate facilities, this problem also addresses the question of when to locate them. We develop the first approximation algorithm for this problem.