Constant Factor Approximation for Capacitated k-Center with Outliers

Abstract

The k-center problem is a classic facility location problem, where given an edge-weighted graph G=(V,E) one is to find a subset of k vertices S, such that each vertex in V is "close" to some vertex in S. The approximation status of this basic problem is well understood, as a simple 2-approximation algorithm is known to be tight. Consequently different extensions were studied.

In the capacitated version of the problem each vertex is assigned a capacity, which is a strict upper bound on the number of clients a facility can serve, when located at this vertex. A constant factor approximation for the capacitated k-center was obtained last year in [Cygan, Hajiaghayi and Khuller, FOCS'12], which was recently improved to a 9-approximation in [An, Bhaskara and Svensson, arXiv'13].

In a different generalization of the problem some clients (denoted as outliers) may be disregarded. Here we are additionally given an integer p and the goal is to serve exactly p clients, which the algorithm is free to choose. In [Charikar et al., SODA'01] the authors presented a 3-approximation for the k-center problem with outliers.

In this paper we consider a common generalization of the two extensions previously studied separately, i.e. we work with the capacitated k-center with outliers. We present the first constant factor approximation algorithm with approximation ratio of 25 even for the case of non-uniform hard capacities.