A Constant Factor Approximation Algorithm for Fault-Tolerant k -Median

Abstract

In this article, we consider the fault-tolerant k -median problem and give the first constant factor approximation algorithm for it. In the fault-tolerant generalization of the classical k -median problem, each client j needs to be assigned to at least r j ⩾ 1 distinct open facilities. The service cost of j is the sum of its distances to the r j facilities, and the k -median constraint restricts the number of open facilities to at most k . Previously, a constant factor was known only for the special case when all r j s are the same, and alogarithmic approximation ratio was known for the general case. In addition, we present the first polynomial time algorithm for the fault-tolerant k -median problem on a path or an HST by showing that the corresponding LP always has an integral optimal solution. We also consider the fault-tolerant facility location problem, in which the service cost of j can be a weighted sum of its distance to the r j facilities. We give a simple constant factor approximation algorithm, generalizing several previous results that work only for nonincreasing weight vectors.