Skip to main content
Log in

When a worse approximation factor gives better performance: a 3-approximation algorithm for the vertex k-center problem

  • Published:
Journal of Heuristics Aims and scope Submit manuscript


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.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Fig. 1
Fig. 2

Similar content being viewed by others


  • Beasly, J.E.: OR-Library: distributing test problems by electronic mail. J. Oper. Res. Soc. 41(11), 1069–1072 (1990)

    Article  Google Scholar 

  • 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)

    Google Scholar 

  • Davidović, T., Ramljak, D., Šelmić, M., Teodorović, D.: Bee colony optimization for the p-center problem. Comput. Oper. Res. 38, 1367–1376 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  • Dyer, M.E., Frieze, A.M.: A simple heuristic for the p-centre problem. Oper. Res. Lett. 3(6), 285–288 (1985)

    Article  MathSciNet  MATH  Google Scholar 

  • Elloumi, S., Labbé, M., Pochet, Y.: A new formulation and resolution method for the p-center problem. INFORMS J. Comput. 16, 84–94 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  • 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)

    Article  Google Scholar 

  • Garey, M., Johnson, D.: Computers and Intractability: A Guide to the Theory of NP-Completeness. W. H. Freeman & Co., New York (1979)

    MATH  Google Scholar 

  • Gonzalez, T.: Clustering to minimize the maximum inter-cluster distance. Theor. Comput. Sci. 38, 293–306 (1985)

    Article  MATH  Google Scholar 

  • Hakimi, S.: Optimum location of switching centers and the absolute centers and medians of a graph. Oper. Res. 12, 450–459 (1964)

    Article  MATH  Google Scholar 

  • Hochbaum, D.: Approximation Algorithms for NP-Hard Problems. PWS Publishing Co., Boston (1997)

    MATH  Google Scholar 

  • Hochbaum, D., Shmoys, D.B.: A best possible heuristic for the k-center problem. Math. Oper. Res. 10, 180–184 (1985)

    Article  MathSciNet  MATH  Google Scholar 

  • Ilhan, T., Pinar, M.Ç.: An efficient exact algorithm for the vertex p-center problem. Optimization online. 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)

    MathSciNet  MATH  Google Scholar 

  • 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)

    Article  Google Scholar 

  • Mihelič, J., Robič, B.: Solving the k-center problem efficiently with a dominating set algorithm. J. Comput. Inf. Technol. 13(3), 225–234 (2005)

    Article  Google Scholar 

  • Minieka, E.: The m-center problem. SIAM Rev. 12, 138–139 (1970)

    Article  MathSciNet  MATH  Google Scholar 

  • Mladenovič, N., Labbé, M., Hansen, P.: Solving the p-center problem with Tabu search and variable neighborhood sSearch. Networks 42, 48–64 (2000)

    Article  MATH  Google Scholar 

  • 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)

    Article  MATH  Google Scholar 

  • Pullan, W.: A memetic genetic algorithm for the vertex p-center problem. Evol. Comput. 16(3), 417–436 (2008)

    Article  Google Scholar 

  • Rana, R., Garg, D.: The analytical study of k-center problem solving techniques. Int. J. Inf. Technol. Knowl. Manag. 1(2), 527–535 (2008)

    Google Scholar 

  • Reinelt, G.: TSPLIB-a traveling salesman problem library. ORSA J. Comput. 3, 376–384 (1991)

    Article  MATH  Google Scholar 

  • 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)

    MATH  Google Scholar 

Download references


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

Correspondence to Jesus Garcia-Diaz.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

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).

Download citation

  • Received:

  • Revised:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: