×

A two-level metaheuristic for the all colors shortest path problem. (English) Zbl 1409.90213

Summary: Given an undirected weighted graph, in which each vertex is assigned to a color and one of them is identified as source, in the all-colors shortest path problem we look for a minimum cost shortest path that starts from the source and spans all different colors. The problem is known to be NP-Hard and hard to approximate. In this work we propose a variant of the problem in which the source is unspecified and show the two problems to be computationally equivalent. Furthermore, we propose a mathematical formulation, a compact representation for feasible solutions and a VNS metaheuristic that is based on it. Computational results show the effectiveness of the proposed approach for the two problems.

MSC:

90C35 Programming involving graphs or networks
90C59 Approximation methods and heuristics in mathematical programming
PDFBibTeX XMLCite
Full Text: DOI

References:

[1] Akcan, H.; Evrendilek, C., Complexity of energy efficient localization with the aid of a mobile beacon, IEEE Commun. Lett., 22, 392-395, (2018) · doi:10.1109/LCOMM.2017.2772876
[2] Akçay, MB; Akcan, H.; Evrendilek, C., All colors shortest path problem on trees, J. Heuristics, (2018) · doi:10.1007/s10732-018-9370-4
[3] Bontoux, B.; Artigues, C.; Feillet, D., A memetic algorithm with a large neighborhood crossover operator for the generalized traveling salesman problem, Comput. Oper. Res., 37, 1844-1852, (2010) · Zbl 1188.90263 · doi:10.1016/j.cor.2009.05.004
[4] Can Bilge, Y., Çagatay, D., Genç, B., Sari, M., Akcan, H., Evrendilek, C.: All colors shortest path problem. arXiv:1507.06865
[5] Carrabs, Francesco; Cerulli, Raffaele; Festa, Paola; Laureana, Federica, On the Forward Shortest Path Tour Problem, 529-537, (2017), Cham
[6] Cormen, T.H., Leiserson, C.E., Rivest, R.L., Stein, C.: Introduction to Algorithms, 3rd edn. The MIT Press, Cambridge (2009) · Zbl 1187.68679
[7] Dimitrijević, V.; Šarić, Z., An efficient transformation of the generalized traveling salesman problem into the traveling salesman problem on digraphs, Inf. Sci., 102, 105-110, (1997) · Zbl 0918.90134 · doi:10.1016/S0020-0255(96)00084-9
[8] Dror, M.; Haouari, M.; Chaouachi, J., Generalized spanning trees, Eur. J. Oper. Res., 120, 583-592, (2000) · Zbl 0985.90076 · doi:10.1016/S0377-2217(99)00006-5
[9] Feremans, C.; Labbé, M.; Laporte, G., A comparative analysis of several formulations for the generalized minimum spanning tree problem, Networks, 39, 29-34, (2002) · Zbl 1001.90066 · doi:10.1002/net.10009
[10] Feremans, C.; Labbé, M.; Laporte, G., The generalized minimum spanning tree problem: Polyhedral analysis and branch-and-cut algorithm, Networks, 43, 71-86, (2004) · Zbl 1069.68114 · doi:10.1002/net.10105
[11] Festa, P.; Guerriero, F.; Laganà, D.; Musmanno, R., Solving the shortest path tour problem, Eur. J. Oper. Res., 230, 464-474, (2013) · Zbl 1317.90065 · doi:10.1016/j.ejor.2013.04.029
[12] Fischetti, M.; Salazar González, JJ; Toth, P., The symmetric generalized traveling salesman polytope, Networks, 26, 113-123, (1995) · Zbl 0856.90116 · doi:10.1002/net.3230260206
[13] Fischetti, M.; Salazar González, JJ; Toth, P., A branch-and-cut algorithm for the symmetric generalized traveling salesman problem, Oper. Res., 45, 378-394, (1997) · Zbl 0893.90164 · doi:10.1287/opre.45.3.378
[14] Golden, B.; Raghavan, S.; Stanojević, D., Heuristic search for the generalized minimum spanning tree problem, INFORMS J. Comput., 17, 290-304, (2005) · Zbl 1239.90099 · doi:10.1287/ijoc.1040.0077
[15] Haouari, M.; Chaouachi, J.; Dror, M., Solving the generalized minimum spanning tree problem by a branch-and-bound algorithm, J. Oper. Res. Soc., 56, 382-389, (2005) · Zbl 1104.90055 · doi:10.1057/palgrave.jors.2601821
[16] Hu, B.; Leitner, M.; Raidl, GR, Combining variable neighborhood search with integer linear programming for the generalized minimum spanning tree problem, J. Heuristics, 14, 473-499, (2008) · Zbl 1211.90309 · doi:10.1007/s10732-007-9047-x
[17] Ihler, E.; Reich, G.; Widmayer, P., Class steiner trees and vlsi-design, Discrete Appl. Math., 90, 173-194, (1999) · Zbl 0921.68054 · doi:10.1016/S0166-218X(98)00090-0
[18] Laporte, G.; Mercure, H.; Nobert, Y., Generalized travelling salesman problem through n sets of nodes: the asymmetrical case, Discrete Appl. Math., 18, 185-197, (1987) · Zbl 0633.90087 · doi:10.1016/0166-218X(87)90020-5
[19] Myung, Y-S; Lee, C-H; Tcha, D-W, On the generalized minimum spanning tree problem, Networks, 26, 231-241, (1995) · Zbl 0856.90117 · doi:10.1002/net.3230260407
[20] Öncan, T.; Cordeau, J-F; Laporte, G., A tabu search heuristic for the generalized minimum spanning tree problem, Eur. J. Oper. Res., 191, 306-319, (2008) · Zbl 1149.90068 · doi:10.1016/j.ejor.2007.08.021
[21] Pop, PC; Kern, W.; Still, G., A new relaxation method for the generalized minimum spanning tree problem, Eur. J. Oper. Res., 170, 900-908, (2006) · Zbl 1091.90068 · doi:10.1016/j.ejor.2004.07.058
[22] Shi, XH; Liang, YC; Lee, HP; Lu, C.; Wang, QX, Particle swarm optimization-based algorithms for TSP and generalized TSP, Inf. Process. Lett., 103, 169-176, (2007) · Zbl 1187.90238 · doi:10.1016/j.ipl.2007.03.010
[23] Snyder, LV; Daskin, MS, A random-key genetic algorithm for the generalized traveling salesman problem, Eur. J. Oper. Res., 174, 38-53, (2006) · Zbl 1116.90091 · doi:10.1016/j.ejor.2004.09.057
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.