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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
##### Keywords:
$$k$$-center; greedy algorithm; approximation algorithm
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##### References:
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