The k-centrum multi-facility location problem

The most common problems studied in network location theory are the p-center and the p-median problems. In the p-center problem the objective is to locate p service facilities to minimize the maximum of the service distances of the n customers to their respective nearest service facility, and in the p-median model the objective is to minimize the sum of these n service distances. (A customer is served only by the closest facility.) We study the p-facility k-centrum model that generalizes and unifies the above problems. The objective of this unifying model is to minimize the sum of the k largest service distances. The p-center and the p-median problems correspond to the cases where k=1 and n, respectively. We present polynomial time algorithms for solving the p-facility k-centrum problem on path and tree graphs. These algorithms can be combined with the general approximation algorithms of Bartal (Proceedings of the 30th Annual ACM Symposium on Theory of Computing, 1998, pp. 161-168) and Charikar et al. (Proceedings of the 30th Annual ACM Symposium on Theory of Computing, 1998, pp. 114-123) to obtain an O(lognloglogn) approximation for a p-facility k-centrum problem defined on a general network.