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Abstract

The $k$-center problem with triangle inequality is that of placing $k$ center nodes in a weighted undirected graph in which the edge weights obey the triangle inequality, so that the maximum distance of any node to its nearest center is minimized. In this paper, we consider a generalization of this problem where, given a number $p$, we wish to place $k$ centers so as to minimize the maximum distance of any node to its $p\th$ closest center. We consider three different versions of this reliable $k$-center problem depending on which of the nodes can serve as centers and non-centers and derive best possible approximation algorithms for all three versions.