Generalized Center Problems with Outliers

Abstract

We study the ℱ-center problem with outliers: Given a metric space ( X , d ), a general down-closed family ℱ of subsets of X , and a parameter m , we need to locate a subset S ∈ ℱ of centers such that the maximum distance among the closest m points in X to S is minimized. Our main result is a dichotomy theorem . Colloquially, we prove that there is an efficient 3-approximation for the ℱ-center problem with outliers if and only if we can efficiently optimize a poly-bounded linear function over ℱ subject to a partition constraint. One concrete upshot of our result is a polynomial time 3-approximation for the knapsack center problem with outliers for which no (true) approximation algorithm was known.