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We consider a fault-tolerant generalization of the classical uncapacitated facility location problem, where each client j has a requirement that r j distinct facilities serve it, instead of just one. We give a 2.076-approximation algorithm for this problem using LP rounding, which is currently the best-known performance guarantee. Our algorithm exploits primal and dual complementary slackness conditions and is based on clustered randomized rounding. A technical difficulty that we overcome is the presence of terms with negative coefficients in the dual objective function, which makes it difficult to bound the cost in terms of dual variables. For the case where all requirements are the same, we give a primal-dual 1.52-approximation algorithm. We also consider a fault-tolerant version of the k-median problem. In the metric k-median problem, we are given n points in a metric space. We must select k of these to be centers, and then assign each input point j to the selected center that is closest to it. In the fault-tolerant version we want j to be assigned to r j distinct centers. The goal is to select the k centers so as to minimize the sum of assignment costs. The primal-dual algorithm for fault-tolerant facility location with uniform requirements also yields a 4-approximation algorithm for the fault-tolerant k-median problem for this case. This the first constant-factor approximation algorithm for the uniform requirements case. [ABSTRACT FROM AUTHOR]

We consider the uncapacitated facility location problem. In this problem, there is a set of locations at which facilities can be built; a fixed cost fi is incurred if a facility is opened at location i. Furthermore, there is a set of demand locations to be serviced by the opened facilities; if the demand location j is assigned to a facility at location i, then there is an associated service cost proportional to the distance between i and j, cij. The objective is to determine which facilities to open and an assignment of demand points to the opened facilities, so as to minimize the total cost. We assume that the distance function c is symmetric and satisfies the triangle inequality. For this problem we obtain a (1 + 2/ e)-approximation algorithm, where 1 + 2/ e≈ 1.736, which is a significant improvement on the previously known approximation guarantees. The algorithm works by rounding an optimal fractional solution to a linear programming relaxation. Our techniques use properties of optimal solutions to the linear program, randomized rounding, as well as a generalization of the decomposition techniques of Shmoys, Tardos, and Aardal [Proceedings of the 29th ACM Symposium on Theory of Computing, El Paso, TX, 1997, pp. 265-274]. [ABSTRACT FROM AUTHOR]