A pre-processing reduction method for the generalized travelling salesman problem
The generalized travelling salesman problem (GTSP) is a variant of the well-known travelling salesman problem. The cities are in this case spread into clusters and only one city from each one must be visited to make a cyclic tour with a minimal cost. Following this assumption, an instance can be sub...
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| Published in | Operational research Vol. 21; no. 4; pp. 2543 - 2591 |
|---|---|
| Main Authors | , , |
| Format | Journal Article |
| Language | English |
| Published |
Berlin/Heidelberg
Springer Berlin Heidelberg
01.12.2021
Springer Nature B.V |
| Subjects | |
| Online Access | Get full text |
| ISSN | 1109-2858 1866-1505 |
| DOI | 10.1007/s12351-019-00533-w |
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| Abstract | The generalized travelling salesman problem (GTSP) is a variant of the well-known travelling salesman problem. The cities are in this case spread into clusters and only one city from each one must be visited to make a cyclic tour with a minimal cost. Following this assumption, an instance can be subject to dimensionality reduction since removing some cities from each cluster will keep feasible solutions and then will preserve feasibility of the instance. Therefore, we propose a new preprocessing technique which consists of selecting from each cluster the nearest nodes to other clusters and removing the nodes that had never been selected in order to reduce the search space size. The suggested approach is tested on a large set of symmetric instances of different sizes picked from the different benchmarks. The reduction is performed in a negligible runtime, while the reduction rate is up to 98%, which is very competitive compared to the only reduction method for the GTSP we are aware of. State-of-the-art solvers for the GTSP were applied to the reduced instances to evaluate their performances. We show that the reduced instances help these solvers to obtain good solutions in a shorter time but do not guarantee to get the optimal ones, while they provide better solutions in a fixed time budget environment. |
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| AbstractList | The generalized travelling salesman problem (GTSP) is a variant of the well-known travelling salesman problem. The cities are in this case spread into clusters and only one city from each one must be visited to make a cyclic tour with a minimal cost. Following this assumption, an instance can be subject to dimensionality reduction since removing some cities from each cluster will keep feasible solutions and then will preserve feasibility of the instance. Therefore, we propose a new preprocessing technique which consists of selecting from each cluster the nearest nodes to other clusters and removing the nodes that had never been selected in order to reduce the search space size. The suggested approach is tested on a large set of symmetric instances of different sizes picked from the different benchmarks. The reduction is performed in a negligible runtime, while the reduction rate is up to 98%, which is very competitive compared to the only reduction method for the GTSP we are aware of. State-of-the-art solvers for the GTSP were applied to the reduced instances to evaluate their performances. We show that the reduced instances help these solvers to obtain good solutions in a shorter time but do not guarantee to get the optimal ones, while they provide better solutions in a fixed time budget environment. |
| Author | Ahiod, Belaïd El Krari, Mehdi El Benani, Youssef Bouazza |
| Author_xml | – sequence: 1 givenname: Mehdi orcidid: 0000-0001-7936-9430 surname: El Krari fullname: El Krari, Mehdi email: mehdi@elkrari.com organization: LRIT, Associated Unit to CNRST (URAC 29), Faculty of Science, Mohammed V University in Rabat – sequence: 2 givenname: Belaïd surname: Ahiod fullname: Ahiod, Belaïd organization: LRIT, Associated Unit to CNRST (URAC 29), Faculty of Science, Mohammed V University in Rabat – sequence: 3 givenname: Youssef Bouazza surname: El Benani fullname: El Benani, Youssef Bouazza organization: Conception and Systems Laboratory, Faculty of Science, Mohammed V University in Rabat |
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| Cites_doi | 10.1016/j.ipl.2007.03.010 10.1007/978-0-387-48793-9_11 10.1016/0020-0255(93)90133-7 10.1007/978-3-319-53480-0_14 10.1016/j.cor.2009.05.004 10.1016/j.ejor.2012.01.011 10.1007/s11047-009-9111-6 10.1023/A:1009881326671 10.1287/opre.45.3.378 10.1016/j.ejor.2004.09.057 10.1016/S0377-2217(97)00142-2 10.1016/j.cor.2012.10.001 10.1016/j.cor.2017.05.010 10.1007/978-3-540-78604-7_4 10.1057/jors.1996.190 10.1016/S0020-0255(96)00084-9 10.1007/s12532-015-0080-8 10.1287/ijoc.3.4.376 10.1007/978-3-540-78987-1_19 10.1016/0166-218X(87)90020-5 |
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| Copyright | Springer-Verlag GmbH Germany, part of Springer Nature 2019 Springer-Verlag GmbH Germany, part of Springer Nature 2019. |
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| Keywords | Dimensionality reduction Nearest nodes Combinatorial optimization Generalized travelling salesman problem |
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| References | Gutin G, Karapetyan D (2008) Generalized traveling salesman problem reduction algorithms. arXiv preprint arXiv:0804.0735 SmithSLImesonFGLNS: an effective large neighborhood search heuristic for the generalized traveling salesman problemComput Oper Res20178711910.1016/j.cor.2017.05.010 DimitrijevićVŠarićZAn efficient transformation of the generalized traveling salesman problem into the traveling salesman problem on digraphsInf Sci19971021–410511010.1016/S0020-0255(96)00084-9 LaporteGNobertYGeneralized travelling salesman problem through n sets of nodes: an integer programming approachINFOR Inf Syst Oper Res19832116175 KarapetyanDGutinGEfficient local search algorithms for known and new neighborhoods for the generalized traveling salesman problemEur J Oper Res2012219223425110.1016/j.ejor.2012.01.011 El Krari M, Ahiod B, El Benani B (2017) Using cluster barycenters for the generalized traveling salesman problem. Springer, Cham, pp 135–143, ISBN 978-3-319-53480-0. https://doi.org/10.1007/978-3-319-53480-0_14 FodorIKA survey of dimension reduction techniquesCent Appl Sci Comput Lawrence Livermore Natl Lab20029118 FischettiMSalazar GonzalezJJTothPA branch-and-cut algorithm for the symmetric generalized traveling salesman problemOper Res199745337839410.1287/opre.45.3.378 Hu B, Raidl GR (2008) Effective neighborhood structures for the generalized traveling salesman problem. In: van Hemert J, Cotta C (eds) Evolutionary computation in combinatorial optimization. EvoCOP 2008. Lecture Notes in Computer Science, vol 4972. Springer, Berlin, Heidelberg, pp 36–47 RenaudJBoctorFFAn efficient composite heuristic for the symmetric generalized traveling salesman problemEur J Oper Res1998108357158410.1016/S0377-2217(97)00142-2 BontouxBArtiguesCFeilletDA memetic algorithm with a large neighborhood crossover operator for the generalized traveling salesman problemComput Oper Res201037111844185210.1016/j.cor.2009.05.004 Bontoux B (2008) Techniques hybrides de recherche exacte et approche: application des problèmes de transport. Ph.D. thesis, University of Avignon and the Vaucluse Noon CE (1988) The generalized traveling salesman problem. PhD thesis, University of Michigan Henry-LabordereALThe record balancing problem: a dynamic programming solution of a generalized travelling salesman problemRIRO1969B–24349 DrorMosheHaouariMohamedGeneralized steiner problems and other variantsJ Comb Optim20004441543610.1023/A:1009881326671 LaporteGAsef-VaziriASriskandarajahCSome applications of the generalized travelling salesman problemJ Oper Res Soc199647121461146710.1057/jors.1996.190 LaporteGSemetFComputational evaluation of a transformation procedure for the symmetric generalized traveling salesman problemINFOR Inf Syst Oper Res1999372114120 SnyderLVDaskinMSA random-key genetic algorithm for the generalized traveling salesman problemEur J Oper Res20061741385310.1016/j.ejor.2004.09.057 LaporteGMercureHNobertYGeneralized travelling salesman problem through n sets of nodes: the asymmetrical caseDiscrete Appl Math198718218519710.1016/0166-218X(87)90020-5 NoonCEBeanJCAn efficient transformation of the generalized traveling salesman problemINFOR Inf Syst Oper Res19933113944 ReineltGTSPLIB—A traveling salesman problem libraryINFORMS J Comput19913437638410.1287/ijoc.3.4.376 LienY-NMaEWahBW-STransformation of the generalized traveling-salesman problem into the standard traveling-salesman problemInf Sci1993741–217718910.1016/0020-0255(93)90133-7 MestriaMOchiLSde Lima MartinsSGrasp with path relinking for the symmetric euclidean clustered traveling salesman problemComput Oper Res201340123218322910.1016/j.cor.2012.10.001 SilberholzJGoldenBBakerEKJosephAMehrotraATrickMAThe generalized traveling salesman problem: a new genetic algorithm approachExtending the horizons: advances in computing, optimization, and decision technologies2007MABoston16518110.1007/978-0-387-48793-9_11 HelsgaunKSolving the equality generalized traveling salesman problem using the Lin–Kernighan–Helsgaun AlgorithmMath Program Comput20157326928710.1007/s12532-015-0080-8 GutinGKarapetyanDKrasnogorNKrasnogorNNicosiaGPavoneMPeltaDMemetic algorithm for the generalized asymmetric traveling salesman problemNature inspired cooperative strategies for optimization (NICSO 2007), Studies in Computational Intelligence2008Berlin, HeidelbergSpringer19921010.1007/978-3-540-78987-1_19 GutinGKarapetyanDA memetic algorithm for the generalized traveling salesman problemNat Comput201091476010.1007/s11047-009-9111-6 SaksenaJPMathematical model of scheduling clients through welfare angencies: II1967Los AngelesDepartment of Electrical Engineering and Medicine, University of Southern California SrivastavaSSKumarSGargRCSenPGeneralized travelling salesman problem through n sets of nodesCORS J1969797101 ShiXHLiangYCLeeHPLuCWangQXParticle swarm optimization-based algorithms for TSP and generalized TSPInf Process Lett2007103516917610.1016/j.ipl.2007.03.010 IK Fodor (533_CR7) 2002; 9 G Laporte (533_CR19) 1996; 47 G Laporte (533_CR17) 1999; 37 M Fischetti (533_CR6) 1997; 45 D Karapetyan (533_CR15) 2012; 219 533_CR23 G Reinelt (533_CR8) 1991; 3 K Helsgaun (533_CR12) 2015; 7 G Gutin (533_CR10) 2010; 9 533_CR5 533_CR1 LV Snyder (533_CR29) 2006; 174 AL Henry-Labordere (533_CR13) 1969; B–2 V Dimitrijević (533_CR3) 1997; 102 J Renaud (533_CR24) 1998; 108 G Gutin (533_CR11) 2008 533_CR14 JP Saksena (533_CR25) 1967 M Mestria (533_CR21) 2013; 40 B Bontoux (533_CR2) 2010; 37 Moshe Dror (533_CR4) 2000; 4 G Laporte (533_CR16) 1983; 21 SS Srivastava (533_CR30) 1969; 7 533_CR9 Y-N Lien (533_CR20) 1993; 74 J Silberholz (533_CR27) 2007 XH Shi (533_CR26) 2007; 103 CE Noon (533_CR22) 1993; 31 G Laporte (533_CR18) 1987; 18 SL Smith (533_CR28) 2017; 87 |
| References_xml | – reference: LaporteGMercureHNobertYGeneralized travelling salesman problem through n sets of nodes: the asymmetrical caseDiscrete Appl Math198718218519710.1016/0166-218X(87)90020-5 – reference: NoonCEBeanJCAn efficient transformation of the generalized traveling salesman problemINFOR Inf Syst Oper Res19933113944 – reference: El Krari M, Ahiod B, El Benani B (2017) Using cluster barycenters for the generalized traveling salesman problem. Springer, Cham, pp 135–143, ISBN 978-3-319-53480-0. https://doi.org/10.1007/978-3-319-53480-0_14 – reference: Gutin G, Karapetyan D (2008) Generalized traveling salesman problem reduction algorithms. arXiv preprint arXiv:0804.0735 – reference: ReineltGTSPLIB—A traveling salesman problem libraryINFORMS J Comput19913437638410.1287/ijoc.3.4.376 – reference: LaporteGAsef-VaziriASriskandarajahCSome applications of the generalized travelling salesman problemJ Oper Res Soc199647121461146710.1057/jors.1996.190 – reference: Hu B, Raidl GR (2008) Effective neighborhood structures for the generalized traveling salesman problem. In: van Hemert J, Cotta C (eds) Evolutionary computation in combinatorial optimization. EvoCOP 2008. Lecture Notes in Computer Science, vol 4972. 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