A continuous facility location problem and its application to a clustering problem

We consider a new problem, which we denote by Continuous Facility Location (ConFL), and its application to the k-Means Problem. Problem ConFL is a natural extension of the Uncapacitated Facility Location Problem where a facility can be any point in Rq. The proposed algorithms are based on a primal-dual technique for spaces with constant dimensions. For the ConFL Problem, we present algorithms with approximation factors 3 + ε and 1.861 + ε for euclidean distances and 9 + ε for squared euclidean distances. For the k-Means Problem (that is restricted to squared euclidean distance), we present an algorithm with approximation factor 54+ ε. All algorithms have good practical behaviour in small dimensions. Comparisons with known algorithms show that the proposed algorithms have good practical behaviour.

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