Connected k-Center and k-Diameter Clustering
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Motivated by an application from geodesy, we introduce a novel clustering problem which is a k-center (or k-diameter) problem with a side constraint. For the side constraint, we are given an undirected connectivity graph G on the input points, and a clustering is now only feasible if every cluster induces a connected subgraph in G. We call the resulting problems the connected k-center problem and the connected k-diameter problem. We prove several results on the complexity and approximability of these problems. Our main result is an O(log^2k)-approximation algorithm for the connected k-center and the connected k-diameter problem. For Euclidean metrics and metrics with constant doubling dimension, the approximation factor of this algorithm improves to O(1). We also consider the special cases that the connectivity graph is a line or a tree. For the line we give optimal polynomial-time algorithms and for the case that the connectivity graph is a tree, we either give an optimal polynomial-time algorithm or a 2-approximation algorithm for all variants of our model. We complement our upper bounds by several lower bounds.
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