Abstract
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 Ge et al. (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 \(CkC\).
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References
Archer A (2001) Two \(O(log^{*}k)\) approximation algorithms for the asymmetric \(k\)-Center problem. Integer programming and cambinatorial optimization. Lecture Notes in Computer Science 2081:1–14
Bondy JA, Murty SR (1976) Graph theory with applications. Elsevier Science Publishing Co., Inc., New York
Chuzhoy J, Guha S, Halperin E, Khama S, Kortsarz G, Krauthgamer R, Naor J (2005) Asymmetric k-Center is \(\log ^* n\) hard to approximate. J ACM 52(4):538–551
Dyer ME, Frieze AM (1985) A simple heuristic for the \(p\)-center problem. Oper Res Lett 3(6):285–288
Ge R, Ester M, Gao BJ, Hu Z, Bhattacharya B, Ben-Moshe B (2008) Joint cluster analysis of attribute data and delationship data: the connected \(k\)-Center problem, algorithms and applications. ACM Trans Knowl Discov Data 2:7
Hochbaum DS, Shmoys DB (1985) A best possible heuristic for the \(k\)-center problem. Math Oper Res 10(2):180–184
Khuller S, Sussmann YJ (2000) The capacitated \(K\)-Center problem. SIAM J Discrete Math 13(3):403–418
Khuller S, Pless R, Sussmann YJ (2000) Fault tolerant \(K\)-center problems. Theor Comput Sci 242(1–2):237–245
Panigrahy R, Vishwanathan S (1998) An \(O(log^{*}n)\) approximation algorithm for the asymmetric \(p\)-Center problem. J Algorithm 27:259–268
Williamson DP, Shmoys DB (2011) The design of approximation algorithms. Cambridge University Press, Cambridge
Acknowledgments
The authors are indebted to the anonymous referees, whose valuable comments lead to a significant improvements on the presentation of the paper. This work is supported by the National Natural Science Foundation of China (No. 11471005, 11371289).
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Liang, D., Mei, L., Willson, J. et al. A simple greedy approximation algorithm for the minimum connected \(k\)-Center problem. J Comb Optim 31, 1417–1429 (2016). https://doi.org/10.1007/s10878-015-9831-8
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DOI: https://doi.org/10.1007/s10878-015-9831-8