Abstract
The vertex k-center selection problem is a well known NP-Hard minimization problem that can not be solved in polynomial time within a \(\rho < 2\) approximation factor, unless \(P=NP\). Even though there are algorithms that achieve this 2-approximation bound, they perform poorly on most benchmarks compared to some heuristic algorithms. This seems to happen because the 2-approximation algorithms take, at every step, very conservative decisions in order to keep the approximation guarantee. In this paper we propose an algorithm that exploits the same structural properties of the problem that the 2-approximation algorithms use, but in a more relaxed manner. Instead of taking the decision that guarantees a 2-approximation, our algorithm takes the best decision near the one that guarantees the 2-approximation. This results in an algorithm with a worse approximation factor (a 3-approximation), but that outperforms all the previously known approximation algorithms on the most well known benchmarks for the problem, namely, the pmed instances from OR-Lib (Beasly in J Oper Res Soc 41(11):1069–1072, 1990) and some instances from TSP-Lib (Reinelt in ORSA J Comput 3:376–384, 1991). However, the \(O(n^4)\) running time of this algorithm becomes unpractical as the input grows. In order to improve its running time, we modified this algorithm obtaining a \(O(n^2 \log n)\) heuristic that outperforms not only all the previously known approximation algorithms, but all the polynomial heuristics proposed up to date.
Similar content being viewed by others
References
Beasly, J.E.: OR-Library: distributing test problems by electronic mail. J. Oper. Res. Soc. 41(11), 1069–1072 (1990)
Daskin, M.: A new approach to solve the vertex p-center problem to optimality: algorithm and computational results. Commun. Oper. Res. Soc. Jpn. 45(9), 428–436 (2000)
Davidović, T., Ramljak, D., Šelmić, M., Teodorović, D.: Bee colony optimization for the p-center problem. Comput. Oper. Res. 38, 1367–1376 (2011)
Dyer, M.E., Frieze, A.M.: A simple heuristic for the p-centre problem. Oper. Res. Lett. 3(6), 285–288 (1985)
Elloumi, S., Labbé, M., Pochet, Y.: A new formulation and resolution method for the p-center problem. INFORMS J. Comput. 16, 84–94 (2004)
Garcia, J., Menchaca, R., Menchaca, R., Quintero, R.: A structure-driven randomized algorithm for the K-center problem. IEEE Latin America Trans. 13(3), 746–752 (2015)
Garey, M., Johnson, D.: Computers and Intractability: A Guide to the Theory of NP-Completeness. W. H. Freeman & Co., New York (1979)
Gonzalez, T.: Clustering to minimize the maximum inter-cluster distance. Theor. Comput. Sci. 38, 293–306 (1985)
Hakimi, S.: Optimum location of switching centers and the absolute centers and medians of a graph. Oper. Res. 12, 450–459 (1964)
Hochbaum, D.: Approximation Algorithms for NP-Hard Problems. PWS Publishing Co., Boston (1997)
Hochbaum, D., Shmoys, D.B.: A best possible heuristic for the k-center problem. Math. Oper. Res. 10, 180–184 (1985)
Ilhan, T., Pinar, M.Ç.: An efficient exact algorithm for the vertex p-center problem. Optimization online. http://www.optimization-online.org/DB_HTML/2001/09/376.html. Accessed 14 July 2015
Kariv, O., Hakimi, S.L.: An algorithmic approach to network location problems, I: the p-centers. J. Appl. Math. 37(3), 513–538 (1979)
Kaveh, A., Nasr, H.: Solving the conditional and unconditional p-center problem with modified harmony search: a real case study. Sci. Iran. A 18(4), 867–877 (2011)
Mihelič, J., Robič, B.: Solving the k-center problem efficiently with a dominating set algorithm. J. Comput. Inf. Technol. 13(3), 225–234 (2005)
Minieka, E.: The m-center problem. SIAM Rev. 12, 138–139 (1970)
Mladenovič, N., Labbé, M., Hansen, P.: Solving the p-center problem with Tabu search and variable neighborhood sSearch. Networks 42, 48–64 (2000)
Pacheco, J.A., Casado, S.: Solving two location models with few facilities by using a hybrid heuristic: a real health resources case. Comput. Oper. Res. 32(12), 3075–3091 (2005)
Pullan, W.: A memetic genetic algorithm for the vertex p-center problem. Evol. Comput. 16(3), 417–436 (2008)
Rana, R., Garg, D.: The analytical study of k-center problem solving techniques. Int. J. Inf. Technol. Knowl. Manag. 1(2), 527–535 (2008)
Reinelt, G.: TSPLIB-a traveling salesman problem library. ORSA J. Comput. 3, 376–384 (1991)
Shmoys D.B.: Computing near-optimal solutions to combinatorial optimization problems. Technical report, Ithaca, NY 14853 (1995)
Vazirani, V.V.: Approximation Algorithms. Springer, New York (2001)
Acknowledgements
This work was sponsored in part by the University of California MEXUS—CONACyT program under Grant CN 15-1451, by the Mexican National Council for Science and Technology (CONACyT) and by the Mexican National Polytechnic Institute (IPN).
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Garcia-Diaz, J., Sanchez-Hernandez, J., Menchaca-Mendez, R. et al. When a worse approximation factor gives better performance: a 3-approximation algorithm for the vertex k-center problem. J Heuristics 23, 349–366 (2017). https://doi.org/10.1007/s10732-017-9345-x
Received:
Revised:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s10732-017-9345-x