Abstract
This paper is concerned with the k-traveling salesman problem (k-TSP), which is a variation of widely studied traveling salesman problem (TSP). Given a set of n cities including a home city and a fixed value k such that \(1 < k \le n\), this problem seeks a tour of minimum length which starts and ends at the home city and visits k cities (including the home city) exactly out of these n cities. Finding a feasible solution to k-TSP involves finding a subset of k cities including the home city first, and then a circular permutation representing a tour of these k cities. In this paper, we have proposed two multi-start heuristic approaches for the k-TSP. The first approach is based on general variable neighborhood search algorithm (GVNS), whereas the latter approach is a hyper-heuristic (HH) approach. A variable neighborhood descent strategy operating over two neighborhood structures is utilized for doing the local search in the GVNS. As part of the hyper-heuristic, two low level heuristics are considered. To the best of our knowledge, these are the first metaheuristic and hyper-heuristic approaches for the k-TSP. To evaluate the performance of our approaches, a set of benchmark instances is created utilizing instances from TSPLIB. Computational results on these benchmark instances show HH approach to be better than GVNS approach.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Balas, E.: The prize collecting traveling salesman problem. Networks 19(6), 621–636 (1989)
Ausiello, G., Bonifaci, V., Leonardi, S., Marchetti-Spaccamela, A.: Prize-collecting traveling salesman and related problems. Handbook Approx. Algorithms Metaheuristics 40, 1–13 (2007)
Blum, A., Ravi, R., Vempala, S.: A constant-factor approximation algorithm for the k MST problem. In: Proceedings of the 28th Annual ACM Symposium on Theory of Computing, pp. 442–448. ACM (1996)
Garg, N.: A 3-approximation for the minimum tree spanning k vertices. In: Proceedings of 37th Annual Symposium on Foundations of Computer Science, pp. 302–309. IEEE, (1996)
Arora, S., Karakostas, G.: A 2+ \(\varepsilon \) approximation algorithm for the k-MST problem. In: Proceedings of the 11th Annual ACM-SIAM Symposium on Discrete Algorithms, pp. 754–759. Society for Industrial and Applied Mathematics, (2000)
Chaudhuri, K., Godfrey, B., Rao, S., Talwar, K.: Paths, trees, and minimum latency tours. In: Proceedings of 44th Annual IEEE Symposium on Foundations of Computer Science, pp. 36–45. IEEE, (2003)
Garg, N.: Saving an epsilon: a 2-approximation for the k-MST problem in graphs. In: Proceedings of the 37th Annual ACM Symposium on Theory of Computing, pp. 396–402. ACM, (2005)
Blum, A., Chawla, S., Karger, D.R., Lane, T., Meyerson, A., Minkoff, M.: Approximation algorithms for orienteering and discounted-reward TSP. SIAM J. Comput. 37(2), 653–670 (2007)
Mladenović, N., Hansen, P.: Variable neighborhood search. Comput. Oper. Res. 24(11), 1097–1100 (1997)
Hansen, P., Mladenović, N., Moreno Pérez, J.A.: Variable neighbourhood search: methods and applications. Ann. Oper. Res. 175(1), 367–407 (2010)
Mladenović, N., Urošević, D., Hanafi, S., Ilić, A.: A general variable neighborhood search for the one-commodity pickup-and-delivery travelling salesman problem. Eur. J. Oper. Res. 220(1), 270–285 (2012)
Mladenović, N., Todosijević, R., Urošević, D.: Two level general variable neighborhood search for attractive traveling salesman problem. Comput. Oper. Res. 52, 341–348 (2014)
Mladenović, N., Todosijević, R., Urošević, D.: An efficient general variable neighborhood search for large travelling salesman problem with time windows. Yugoslav J. Oper. Res. 23(1), 19–30 (2013)
Todosijević, R., Mjirda, A., Mladenović, M., Hanafi, S., Gendron, B.: A general variable neighborhood search variants for the travelling salesman problem with draft limits. Optim. Lett. 11(6), 1047–1056 (2017)
Venkatesh, P., Srivastava, G., Singh, A.: A general variable neighborhood search algorithm for the k-traveling salesman problem. In: Proceedings of the 8th International Conference on Advances in Computing & Communications (ICACC-2018), Procedia Computer Science, vol. 143, pp. 189–196. Elsevier, (2018)
Burke, E.K., Gendreau, M., Hyde, M., Kendall, G., Ochoa, G., Özcan, E., Qu, R.: Hyper-heuristics: a survey of the state of the art. J. Oper. Res. Soc. 64(12), 1695–1724 (2013)
Denzinger, J., Fuchs, M., Fuchs, M.: High performance ATP systems by combining several AI methods. In: Proceedings of the 15th International Joint Conference on Artifical Intelligence, vol. 1, pp. 102–107, San Francisco, CA, USA. Morgan Kaufmann Publishers Inc (1997)
Fisher, H.: Probabilistic learning combinations of local job-shop scheduling rules. In: Industrial scheduling, pp. 225–251, (1963)
Crowston, W.B., Glover, F., Thompson, G.L., Trawick, J.D.: Probabilistic and parametric learning combinations of local job shop scheduling rules. Technical report, Research Memorandum, No. 117, GSIA, Carnegie Mellon University, Pittsburgh, (1963)
Cowling, P., Kendall, G., Soubeiga, E.: A hyperheuristic approach to scheduling a sales summit. In: International Conference on the Practice and Theory of Automated Timetabling, pp. 176–190. Springer, (2000)
Chakhlevitch, K., Cowling, P.: Hyperheuristics: recent developments. In: Adaptive and multilevel metaheuristics, pp. 3–29. Springer, (2008)
Drake, J.H., Özcan, E., Burke, E.K.: An improved choice function heuristic selection for cross domain heuristic search. In: International Conference on Parallel Problem Solving from Nature, pp. 307–316. Springer, (2012)
Kendall, G., Li, J.: Competitive travelling salesmen problem: a hyper-heuristic approach. J. Oper. Res. Soc. 64(2), 208–216 (2013)
Pandiri, V., Singh, A.: A hyper-heuristic based artificial bee colony algorithm for k-interconnected multi-depot multi-traveling salesman problem. Inf. Sci. 463, 261–281 (2018)
Zalilah Abd Aziz: Ant colony hyper-heuristics for travelling salesman problem. Procedia Comput. Sci. 76, 534–538 (2015)
Choong, S.S., Wong, L.-P., Lim, C.P.: An artificial bee colony algorithm with a modified choice function for the traveling salesman problem. Swarm Evol. Comput. 44, 622–635 (2019)
Wilcoxon, F., Katti, S.K., Wilcox, R.A.: Critical values and probability levels for the wilcoxon rank sum test and the wilcoxon signed rank test. Sel. Tab. Math. Stat. 1, 171–259 (1970)
Acknowledgements
The authors would like to express their sincere gratitude to two anonymous reviewers and the editor for their valuable comments and suggestions which helped in improving the quality of this manuscript. The first author is grateful to the Council of Scientific and Industrial Research (CSIR), Government of India for supporting his research through a Senior Research Fellowship. The second author acknowledges the support of the research Grant No. MTR/2017/000391 received under MATRICS scheme from Science and Engineering Research Board (SERB), Government of India.
Author information
Authors and Affiliations
Corresponding author
Additional information
Publisher's Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
Appendices
Appendix I
This appendix presents the results obtained by our approaches on each TSPLIB instance under each of the six groups. Therefore, there are six tables, each corresponding to a group. Tables 4, 5 and 6 correspond to groups (Small, SR), (Medium, SR) and (Large, SR) respectively. These three tables report the performance of GVNS, HH_RAND and HH_GREEDY under short run. On the other hand, tables 7, 8 and 9 correspond to groups (Small, LR), (Medium, LR) and (Large, LR) respectively, and report the performance of our approaches under long run. In all these tables, the first column lists the instance names. The numerical values at the end of an instance name specifies the number of cities in the corresponding instance. The second column (k) reports the number of cities (including the home city) that the salesman has to visit. The columns Best, Worst & Average under an approach reports the best, worst and average solution quality obtained over ten independent runs respectively by that approach. The best values are reported in bold font for ease of identification. Please note that we have also reported the worst solution obtained by various approaches over ten runs here. The knowledge about the worst solution obtained aids in ascertaining the robustness of an approach in comparison to other approaches. As can be seen from these tables, even the worst solution obtained by HH_RAND is better than the worst solution obtained by other two approaches in most of the cases.
Appendix II
This appendix presents the convergence behavior of our approaches. For studying the convergence behavior, three instances of different sizes, namely gr229, ali535 and p654, have been considered. Figures 3, 4 and 5 plot the convergence behavior of our three approaches, viz. GVNS, HH_RAND and HH_GREEDY respectively for the instances gr229, ali535 and p654 under small, medium and large scenarios. The convergence graphs depict that both HH_RAND and HH_GREEDY converges faster than the GVNS. When it comes to the comparison between HH_RAND and HH_GREEDY based on their convergence behavior, there is only a minute difference in their convergence behavior except for the large scenario of p654, where HH_RAND clearly converges faster than HH_GREEDY.
Convergence behavior on instance gr229 under different scenarios
Convergence behavior on instance ali535 under different scenarios
Convergence behavior on instance p654 under different scenarios
Rights and permissions
About this article
Cite this article
Pandiri, V., Singh, A. Two multi-start heuristics for the k-traveling salesman problem. OPSEARCH 57, 1164–1204 (2020). https://doi.org/10.1007/s12597-020-00463-8
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s12597-020-00463-8