Approximation algorithms for facility location problems

We present new approximation algorithms for several facility location problems. In each facility location problem that we study, there is a set of locations at which we may build a facility (such as a warehouse), where the cost of building at location i is fi; furthermore, there is a set of client locations (such as stores) that require to be serviced by a facility, and if a client at location j is assigned to a facility at location i, a cost of cij is incurred that is proportional to the distance between i and j. The objective is to determine a set of locations at which to open facilities so as to minimize the total facility and assignment costs. In the uncapacitated case, each facility can service an unlimited number of clients, whereas in the capacitated case, each facility can serve, for example, at most u clients. These models and a number of closely related ones have been studied extensively in the Operations Research literature. We shall consider the case in which the distances between locations are non-negative, symmetric and satisfy the triangle inequality. For the uncapacitated facility location, we give a polynomial-time algorithm that finds a solution of cost within a factor of 3.16 of the optimal. This is the first constant performance guarantee known for this problem. We also present approximation algorithms with constant performance guarantees for a number of capacitated models as well as a generalization in which there is a 2-level hierarchy of facilities. Our results are based on the filtering and rounding technique of Lin & Vitter. We also give a randomized variant of this technique that can then be derandomized to yield improved deterministic performance guarantees. ∗ [email protected]. School of Operations Research & Industrial Engineering and Department of Computer Science, Cornell University, Ithaca, NY 14853. Research partially supported by NSF grants CCR-9307391 and DMS9505155 and ONR grant N00014-96-1-0050O. † [email protected]. Department of Computer Science and School of Operations Research & Industrial Engineering, Cornell University, Ithaca, NY 14853. Research partially supported by NSF grants DMI-9157199 and DMS-9505155 and ONR grant N00014-96-1-0050O. [email protected]. Department of Computer Science, Utrecht University, Utrecht, The Netherlands. Research partially supported by NSF grant CCR-9307391, and by ESPRIT Long Term Research Project No. 20244 (project ALCOM-IT: Algorithms and Complexity in Information Technology).