A branch-and-cut algorithm for the maximum covering cycle problem

  • Eduardo Álvarez-Miranda
  • Markus Sinnl
S.I.: Decomposition Methods for Hard Optimization Problems


In many applications, such as telecommunications and routing, we seek for cost-effective infrastructure or operating layouts so that many nodes (e.g., customers) of a support network (typically modeled by a graph) are covered by, or at least are easily reachable from, such a layout. In this paper, we study the maximum covering cycle problem. In this problem we are given a non-complete graph, and the goal is to find a cycle, such that the number of nodes which are either on the cycle or are adjacent to the cycle is maximized. We design a branch-and-cut framework for solving the problem. The framework contains valid inequalities, lifted inequalities and a primal heuristic. In a computational study, we compare our framework to previous work available for this problem. The results reveal that our approach significantly outperforms the previous approach. In particular, all available instances from the literature could be solved to optimality with our approach, most of them within a few seconds.


Covering problems Branch-and-cut Optimal cycle problems Domination problems 



E. Álvarez-Miranda acknowledges the support of the Chilean Council of Scientific and Technological Research, CONICYT, through the FONDECYT Grant N.1180670 and through the Complex Engineering Systems Institute (ICM-FIC:P-05-004-F, CONICYT:FB0816). The research of M. Sinnl was supported by the Austrian Research Fund (FWF, Project P 26755-N19).


  1. Aazami, A. (2010). Domination in graphs with bounded propagation: Algorithms, formulations and hardness results. Journal of combinatorial optimization, 19(4), 429–456.CrossRefGoogle Scholar
  2. Arkin, E., & Hassin, R. (1994). Approximation algorithms for the geometric covering salesman problem. Discrete Applied Mathematics, 55(3), 197–218.CrossRefGoogle Scholar
  3. Balas, E. (1989). The asymmetric assignment problem and some new facets of the traveling salesman polytope on a directed graph. SIAM Journal on Discrete Mathematics, 2(4), 425–451.CrossRefGoogle Scholar
  4. Bley, A., Ljubić, I., & Maurer, O. (2017). A node-based ilp formulation for the node-weighted dominating steiner problem. Networks, 69(1), 33–51.CrossRefGoogle Scholar
  5. Colbourn, C., & Stewart, L. (1991). Permutation graphs: Connected domination and Steiner trees. In S. Hedetniemi (Ed.), Topics on Domination (Vol. 48, pp. 179–189)., Annals of Discrete Mathematics New York: Elsevier.CrossRefGoogle Scholar
  6. Current, J., & Schilling, D. (1989). The covering salesman problem. Transportation Science, 23(3), 208–213.CrossRefGoogle Scholar
  7. Current, J., & Schilling, D. (1994). The median tour and maximal covering tour problems: Formulations and heuristics. European Journal of Operational Research, 73(1), 114–126.CrossRefGoogle Scholar
  8. Dantzig, G., Fulkerson, R., & Johnson, S. (1954). Solution of a large-scale traveling-salesman problem. Journal of the Operations Research Society of America, 2(4), 393–410.CrossRefGoogle Scholar
  9. Fischetti, M., Salazar-González, J.-J., & Toth, P. (1997). A branch-and-cut algorithm for the symmetric generalized traveling salesman problem. Operations Research, 45(3), 378–394.CrossRefGoogle Scholar
  10. Fischetti, M., Salazar-González, J., & Toth, P. (1999). Solving the orienteering problem through branch-and-cut. INFORMS Journal on Computing, 10, 133–148.CrossRefGoogle Scholar
  11. Fischetti, M., Leitner, M., Ljubić, I., Luipersbeck, M., Monaci, M., Resch, M., et al. (2017). Thinning out steiner trees: A node-based model for uniform edge costs. Mathematical Programming Computation, 9(2), 203–229.CrossRefGoogle Scholar
  12. Gendreau, M., Laporte, G., & Semet, F. (1997). The covering tour problem. Operations Research, 45(4), 568–576.CrossRefGoogle Scholar
  13. Gendron, B., Lucena, A., da Cunha, A., & Simonetti, L. (2014). Benders decomposition, branch-and-cut, and hybrid algorithms for the minimum connected dominating set problem. INFORMS Journal on Computing, 26(4), 645–657.CrossRefGoogle Scholar
  14. Golden, B., Naji-Azimi, Z., Raghavan, S., Salari, M., & Toth, P. (2012). The generalized covering salesman problem. INFORMS Journal on Computing, 24(4), 534–553.CrossRefGoogle Scholar
  15. Gollowitzer, S., & Ljubić, I. (2011). Mip models for connected facility location: A theoretical and computational study. Computers & Operations Research, 38(2), 435–449.CrossRefGoogle Scholar
  16. Grosso, A., Salassa, F., & Vancroonenburg, W. (2016). Searching for a cycle with maximum coverage in undirected graphs. Optimization Letters, 10(7), 1493–1504.CrossRefGoogle Scholar
  17. Haynes, T., Hedetniemi, S., & Slater, P. (1998). Fundamentals of domination in graphs (1st ed.)., Pure and applied mathematics Boca Raton: CRC Press.Google Scholar
  18. Hoffman, K., Padberg, M., & Rinaldi, G. (2013). Traveling salesman problem. Encyclopedia of operations research and management science (pp. 1573–1578). Berlin: Springer.CrossRefGoogle Scholar
  19. Jeong, I. (2017). An optimal approach for a set covering version of the refueling-station location problem and its application to a diffusion model. International Journal of Sustainable Transportation, 11(2), 86–97.CrossRefGoogle Scholar
  20. Jozefowiez, N., Semet, F., & Talbi, E. (2007). The bi-objective covering tour problem. Computers & Operations research, 34(7), 1929–1942.CrossRefGoogle Scholar
  21. Koch, T., & Martin, A. (1998). Solving steiner tree problems in graphs to optimality. Networks, 32(3), 207–232.CrossRefGoogle Scholar
  22. Koch, T., Martin, A., & Voß, S. (2001). SteinLib: An updated library on Steiner tree problems in graphs. Steiner trees in industry, 11, 285–326.CrossRefGoogle Scholar
  23. Kratochv, J., Proskurowski, A., & Telle, J. (1998). Complexity of graph covering problems. Nordic Journal of Computing, 5, 173–195.Google Scholar
  24. Kruskal, J. (1956). On the shortest spanning subtree of a graph and the traveling salesman problem. Proceedings of the American Mathematical Society, 7(1), 48–50.CrossRefGoogle Scholar
  25. Leitner, M., Ljubić, I., Salazar-González, J.-J., & Sinnl, M. (2017). An algorithmic framework for the exact solution of tree-star problems. European Journal of Operational Research, 261(1), 54–66.CrossRefGoogle Scholar
  26. Ozbaygin, G., Yaman, H., & Karasan, O. (2016). Time constrained maximal covering salesman problem with weighted demands and partial coverage. Computers & Operations Research, 76, 226–237.CrossRefGoogle Scholar
  27. Shaelaie, M., Salari, M., & Naji-Azimi, Z. (2014). The generalized covering traveling salesman problem. Applied Soft Computing, 24, 867–878.CrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media, LLC, part of Springer Nature 2018

Authors and Affiliations

  1. 1.Department of Industrial EngineeringUniversidad de TalcaCuricóChile
  2. 2.Department for Statistics and Operations ResearchUniversity of ViennaViennaAustria

Personalised recommendations