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Evolutionary operators for the Hamiltonian completion problem

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Abstract

This paper deals with evolutionary algorithms for solving the Hamiltonian completion problem. More precisely, the paper is concerned with a collection of crossover and mutation operators, which mostly originate from the traveling salesman problem, but have further on been modified or customized for Hamiltonian completion. The considered crossovers and mutations are tested on a set of randomly generated problem instances. The obtained experimental results clearly show that the behavior and relative ranking of the operators within the Hamiltonian completion environment are different than within the traveling salesman environment. Moreover, it is shown that our modified or custom-designed operator variants accomplish much better results for Hamiltonian completion than the standard variants.

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References

  • Agnetis A, Detti P, Meloni C, Pacciarelli D (2001) A linear algorithm for the Hamiltonian completion number of the line graph of a tree. Inf Process Lett 79:17–24

    Article  MathSciNet  Google Scholar 

  • Ahmed ZH (2014) Improved genetic algorithms for the traveling salesman problem. Int J Process Manag Benchmark 4:109–124

    Article  Google Scholar 

  • Bienkowski M, Zalewski P (2013) (1,2)-Hamiltonian completion on a matching. Int J Found Comput S 24:95–108

    Article  MathSciNet  Google Scholar 

  • Choi I, Kim S, Kim H (2003) A genetic algorithm with a mixed region search for the asymmetric traveling salesman problem. Comput Oper Res 30:773–786

    Article  MathSciNet  Google Scholar 

  • Contreras-Bolton C, Parada V (2015) Automatic combination of operators in a genetic algorithm to solve the traveling salesman problem. PLoS ONE. https://doi.org/10.1371/journal.pone.0137724

    Article  Google Scholar 

  • Cook WJ (2012) In pursuit of the traveling salesman: mathematics at the limits of computation. Princeton University Press, Princeton

    MATH  Google Scholar 

  • Cook WJ (2015) Concorde TSP solver. University od Waterloo, Ontario, Canada. http://www.math.uwaterloo.ca/tsp/concorde/index.html. Accessed 5 January 2020

  • Detti P, Meloni C (2004) A linear algorithm for the Hamiltonian completion number of the line graph of a cactus. Discrete Appl Math 136:197–215

    Article  MathSciNet  Google Scholar 

  • Detti P, Meloni C, Pranzo M (2007) Local search algorithms for finding the Hamiltonian completion number of line graphs. Ann Oper Res 156:5–24

    Article  MathSciNet  Google Scholar 

  • Detti P, Meloni C, Pranzo M (2013) A lower bound on the Hamiltonian path completion number of a line graph. Appl Math Comput 220:296–304

    MathSciNet  MATH  Google Scholar 

  • Eiben AE, Smith JE (2015) Introduction to evolutionary computing. Natural computing series, 2nd edn. Springer, Berlin

    Book  Google Scholar 

  • Franzblau DS, Raychaudhuri A (2002) Optimal Hamiltonian completions and path covers for trees, and a reduction to maximum flow. ANZIAM J 44:193–204

    Article  MathSciNet  Google Scholar 

  • Gamarnik D, Sviridenko M (2005) Hamiltonian completions of sparse random graphs. Discrete Appl Math 152:139–158

    Article  MathSciNet  Google Scholar 

  • Garey MR, Johnson DS (1979) Computers and intractability: a guide to the theory of NP-completness. W.H Freeman, San Francisco

    MATH  Google Scholar 

  • Gog A, Chira C (2011) Comparative analysis of recombination operators in genetic algorithms for the traveling salesman problem. In: Corchado E, Kurzynski M, Wozniak M (eds) Hybrid artificial intelligent systems—6th international conference, HAIS 2011, Wroclaw, Poland, May 23–25, 2011, Proceedings part II. LNCS vol 6679. Springer, Berlin, pp 10–17

  • Grefenstette JJ, Gopal R, Rosmaita BJ, Van Gucht D (1985) Genetic algorithms for the traveling salesman problem. In: Grefenstette JJ (ed) Proceedings of the 1st international conference on genetic algorithms, Pittsburgh PA, July 1985. Lawrence Erlbaum Associates, Mahwah NJ, pp 160–168

  • Gutin G, Punnen AP (2007) The traveling salesman problem and its variations. Springer, New York

    Book  Google Scholar 

  • Helsgaun K (2018) LKH Version 2.0.9. Roskilde University, Denmark. http://akira.ruc.dk/ keld/research/LKH. Accessed 5 January 2020

  • Jungnickel D (2013) Graphs, networks and algorithms, 4th edn. Springer, Berlin

    Book  Google Scholar 

  • Komlos J, Szeweredi E (2006) Limit distribution for the existence of Hamiltonian cycles in a random graph. Discrete Math 306:1032–1038

    Article  Google Scholar 

  • Korneyenko NM (1994) Combinatorial algorithms on a class of graphs. Discrete Appl Math 54:215–217

    Article  MathSciNet  Google Scholar 

  • Korte B, Wygen J (2012) Combinatorial optimization—theory and algorithms, 5th edn. Springer, Berlin

    Book  Google Scholar 

  • Larranaga P, Kuijpers CMH, Murga RH, Inza I, Dizdarevic S (1999) Genetic algorithms for the travelling salesman problem: a review of representations and operators. Artif Intell Rev 13:129–170

    Article  Google Scholar 

  • Li C, Zhan W, Xu J, Yang L (2011) Development of two assistant crossover operators to solve the traveling salesman problem. Int J Adv Comput Technol 3:178–184

    Google Scholar 

  • Maretić HP, Grbić A (2015) A heuristic approach to Hamiltonian completion problem (HCP). In: Biljanović P (ed) Proceedings of the 38th international convention on information and communication technology, electronics and microelectronics, MIPRO 2015, Opatija, Croatia, May 25-29, 2015. MIPRO Society, Rijeka, Croatia, pp 1607-1612

  • Mchedlidze T, Symvonis A (2009) Crossing-optimal acyclic Hamiltonian path completion and its application to upward topological book embeddings. In: Das S, Uchara R (eds) Proceedings of the third international workshop on algorithms and computation, WALCOM 2009, Kolkata, India, February 18–20 (2009) LNCS, vol 5431. Springer, Berlin, pp 250–261

  • Mchedlidze T, Symvonis A (2009) Crossing-free acyclic Hamiltonian path completion for planar st-digraphs. In: Dong Y, Du D-Z, Ibarra O (eds) Proceedings of the 20th international symposium on algorithms and computation, ISAAC 2009, Honolulu, Hawaii, USA, December 16–18 (2009) LNCS, vol 5878. Springer, Berlin, pp 882–891

  • Meloni C, Naso D, Turchiano B (2006) Setup coordination between two stages of a production system: a multi-objective evolutionary approach. Ann Oper Res 147:175–198

    Article  MathSciNet  Google Scholar 

  • Michalewicz Z (1996) Genetic algorithms + data structures = evolution programs, 3rd edn. Springer, New York

    Book  Google Scholar 

  • Nagata Y, Kobayashi S (2013) A powerful genetic algorithm using edge assembly crossover for the traveling salesman problem. INFORMS J Comput 25:346–363

    Article  MathSciNet  Google Scholar 

  • Naso D, Turchiano B, Meloni C (2006) Single and multi-objective evolutionary algorithms for the coordination of serial manufacturing operations. J Intell Manuf 17:251–270

    Article  Google Scholar 

  • Nikolopolous SD (2004) Parallel algorithms for Hamiltonian problems on quasi-threshold graphs. J Parallel Distrib Comput 64:48–67

    Article  Google Scholar 

  • Papadimitrou CH, Steiglitz K (1998) Combinatorial optimization: algorithms and complexity. Dover Publications, Mineola

    Google Scholar 

  • Pongcharoen P, Stewardson DJ, Hicks C, Braiden PM (2001) Applying designed experiments to optimize the performance of genetic algorithms used for scheduling complex products in the capital goods industry. J Appl Stat 28:441–455

    Article  MathSciNet  Google Scholar 

  • Raychaudhuri A (1995) The total interval number of a tree and the Hamiltonian completion number of its line graph. Inf Process Lett 56:299–306

    Article  MathSciNet  Google Scholar 

  • Raychaudhuri A (1999) On Hamiltonian path completion number of an interval graph. In: Stanton RG, Allston JL (eds) Proceedings of the 30th southeastern international conference on combinatorics, graph theory and computing, Boca Raton, FL, March 8–12, 1999. Utilitas Mathematica Publishing, Winnipeg, Manitoba, Canada, pp 193–198

  • Spiegel M, Stephens L (2014) Schaum’s outline of statistics, 5th edn. McGraw-Hill, New York

    Google Scholar 

  • Starkweather T, McDaniel S, Mathias KE, Whitley LD, Whitley C (1991) A comparison of genetic sequencing operators. In: Belew RK, Booker LB (eds) Proceedings of the 4th international conference on genetic algorithms, San Diego CA, July 1991. Morgan Kaufmann, San Francisco CA, pp 69–76

  • Tsai HK, Yang J-M, Tsai Y-F, Kao C-Y (2004) An evolutionary algorithm for large traveling salesman problems. IEEE Trans Syst Man Cybern B Cybern 34:1718–1729

    Article  Google Scholar 

  • Wang J, Ersoy OK, He M, Wang F (2016) Multi-offspring genetic algorithm and its application to the traveling salesman problem. Appl Soft Comput 43:415–423

    Article  Google Scholar 

  • Wang J, Yang W (2019) The Turán number for spanning linear forests. Discrete Appl Math 254:291–294

    Article  MathSciNet  Google Scholar 

  • Wu Q, Lu CL, Lee RC-T (2005) The approximability of the weighted Hamiltonian path completion problem on a tree. Theor Comput Sci 341:385–397

    Article  MathSciNet  Google Scholar 

  • Zhao Z, Chen Y, Zhao Z (2016) A study on the traveling salesman problem using genetic algorithms. Int J Simul Syst Sci Technol 17:1–6

    Google Scholar 

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Acknowledgements

This work has been fully supported by Croatian Science Foundation under the Project IP-2018-01-5591. The authors would like to thank the reviewers for their useful remarks on an earlier version of the paper.

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Authors and Affiliations

  1. Faculty of Economics and Business, University of Zagreb, Trg J.F. Kennedyja 6, 10000, Zagreb, Croatia

    Krunoslav Puljić

  2. Department of Mathematics, Faculty of Science, University of Zagreb, Bijenička cesta 30, 10000, Zagreb, Croatia

    Robert Manger

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  1. Krunoslav Puljić
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  2. Robert Manger
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Correspondence to Robert Manger.

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Puljić, K., Manger, R. Evolutionary operators for the Hamiltonian completion problem. Soft Comput 24, 18073–18088 (2020). https://doi.org/10.1007/s00500-020-05063-8

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