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A simple greedy approximation algorithm for the minimum connected \(k\)-center problem. (English) Zbl 1347.90073
Summary: In this paper, we consider the connected \(k\)-Center (\(CkC\)) problem, which can be seen as the classic \(k\)-Center problem with the constraint of internal connectedness, i.e., two nodes in a cluster are required to be connected by an internal path in the same cluster. \(CkC\) was first introduced by R. Ge et al. [“Joint cluster analysis of attribute data and relationship data; the connected \(k\)-center problem, algorithms and applications”, ACM Trans. Knowl. Discov. Data 2, 7 (2008)], in which they showed the \(NP\)-completeness for this problem and claimed a polynomial time approximation algorithm for it. However, the running time of their algorithm might not be polynomial, as one key step of their algorithm involves the computation of an \(NP\)-hard problem. We first present a simple polynomial time greedy-based \(2\)-approximation algorithm for the relaxation of \(CkC\) – the \(CkC^*\). Further, we give a \(6\)-approximation algorithm for \(CkC\).
MSC:
90C27 Combinatorial optimization
90C59 Approximation methods and heuristics in mathematical programming
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