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Modeling and solving the bi-objective minimum diameter-cost spanning tree problem

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Abstract

The bi-objective minimum diameter-cost spanning tree problem (bi-MDCST) seeks spanning trees with minimum total cost and minimum diameter. The bi-objective version generalizes the well-known bounded diameter minimum spanning tree problem. The bi-MDCST is a NP-hard problem and models several practical applications in transportation and network design. We propose a bi-objective multiflow formulation for the problem and effective multi-objective metaheuristics: a multi-objective evolutionary algorithm and a fast nondominated sorting genetic algorithm. Some guidelines on how to optimize the problem whenever a priority order can be established between the two objectives are provided. In addition, we present bi-MDCST polynomial cases and theoretical bounds on the search space. Results are reported for four representative test sets.

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References

  1. Arroyo, J.E.C., Vieira, P.S., Vianna, D.S.: A GRASP algorithm for the multi-criteria minimum spanning tree problem. Ann. Oper. Res. 159, 125–133 (2008)

    Article  Google Scholar 

  2. Bui, M., Butelle, F., Lavault, C.: A distributed algorithm for constructing the minimum spanning tree problem. J. Parallel Distrib. Comput. Arch. 64, 571–577 (2004)

    Article  Google Scholar 

  3. Carrano, E.G., Fonseca, C.M., Takahashi, R.H.C., Pimenta, L.C.A., Neto, O.M.: A preliminary comparison of tree encoding schemes for evolutionary algorithms, pp. 1969–1974. Montreal, Canadá, (2007)

  4. Chinchuluun, A., Pardalos, P.M.: A survey of recent developments in multiobjective optimization. Ann. Oper. Res. 154, 29–50 (2007)

    Article  Google Scholar 

  5. Coello, C.A.C., Lamont, G.B., Veldhuizen, D.A.V.: Evolutionary Algorithms for Solving Multi-Objective Problems (Genetic and Evolutionary Computation). Springer, New York (2006)

    Google Scholar 

  6. Cormen, T.H., Leiserson, C.E., Rivest, R., Stein, C.: Introduction to Algorithms, 3rd edn. The MIT Press, Cambridge (2009)

    Google Scholar 

  7. Deb, K., Pratap, A., Agarwal, S., Meyarivan, T.: A fast and elitist multiobjective genetic algorithm: NSGA-II. IEEE Trans. Evol. Comput. 6(2), 182–197 (2002)

    Article  Google Scholar 

  8. Ehrgott, M., Gandibleux, X.: A survey and annotated bibliography of multiobjective combinatorial optimization. OR-Spektrum 22, 425–460 (2000)

    Article  Google Scholar 

  9. Garey, M.R., Johnson, D.S.: Computers and Intractability: A Guide to the Theory of NP-Completeness. W.H. Freeman, New York (1979)

    Google Scholar 

  10. Gouveia, L., Magnanti, T.L.: Network flow models for designing diameter-constrained minimum-spanning and Steiner trees. Networks 41, 159–173 (2003)

    Article  Google Scholar 

  11. Gouveia, L., Simonetti, L., Uchoa, E.: Modeling hop-constrained and diameter-constrained minimum spanning tree problems as Steiner tree problems over layered graphs. Math. Program. 128, 123–148 (2011)

    Article  Google Scholar 

  12. Handler, G.Y.: Minimax location of a facility in an undirected graph. Transp. Sci. 7, 287–293 (1978)

    Article  Google Scholar 

  13. Hassin, R., Tamir, A.: On the minimum diameter spanning tree problem. Inf. Process. Lett. 53(2), 109–111 (1995)

    Article  Google Scholar 

  14. Ho, J.-M., Lee, D.T., Chang, C.-H., Wong, K.: Minimum diameter spanning trees and related problems. SIAM J. Comput. 20(5), 987–997 (1991)

    Article  Google Scholar 

  15. Kumar, R., Singh, P.K., Chakrabarti, P.P.: Multiobjective EA approach for improved quality of solutions for spanning tree problem. In: Proceedings International Conference on Evolutionary Multi-Criterion Optimization (EMO), Lecture Notes in Computer Science, pp. 811–825. Springer (2005)

  16. Kumar, R., Singh, P.K.: On quality performance of heuristic and evolutionary algorithms for biobjective minimum spanning trees. In: Proceedings of the 9th Annual Conference on Genetic and Evolutionary Computation, GECCO ’07, pp. 2259–2259. New York, NY, USA. ACM (2007)

  17. Kumar, R., Rockett, P.I.: Improved sampling of the Pareto-front in multiobjective genetic optimizations by steady-state evolution: a Pareto converging genetic algorithm. Evol. Comput. 10(3), 283–314 (2002)

    Article  Google Scholar 

  18. Lemesre, J., Dhaenens, C., Talbi, E.G.: Parallel Partitioning Method (PPM): a new exact method to solve bi-objective problems. Comput. Oper. Res. 34(8), 2450–2462 (2007)

    Google Scholar 

  19. Lucena, A., Ribeiro, C., Santos, A.C.: A hybrid heuristic for the diameter constrained minimum spanning tree problem. J. Global Optim. 46, 363–381 (2010)

    Article  Google Scholar 

  20. Marathe, M.V., Ravi, R., Sundaram, R., Ravi, S.S., Rosenkrantz, D.J., Hunt III, H.B.: Bicriteria network design problems. J. Algorithms 28(1), 142–171 (1998)

    Article  Google Scholar 

  21. Neumann, F., Wegener, I.: Minimum spanning trees made easier via multi-objective optimization. In: Proceedings of the 2005 Conference on Genetic and Evolutionary Computation, GECCO ’05, pp. 763–769, New York, NY, USA. ACM (2005)

  22. Noronha, T.F., Santos, A.C., Ribeiro, C.C.: Constraint programming for the diameter constrained minimum spanning tree problem. Electron. Notes Discrete Math. 30, 93–98 (2008)

    Article  Google Scholar 

  23. Noronha, T.F., Ribeiro, C.C., Santos, A.C.: Solving diameter-constrained minimum spanning tree problems by constraint programming. Int. Trans. Oper. Res. 17(5), 653–665 (2010)

    Article  Google Scholar 

  24. Raidl, G.R., Julstrom, B.A.: Greedy heuristics and an evolutionary algorithm for the bounded-diameter minimum spanning tree problem. In: Proceedings of the 18th ACM Symposium on Applied Computing, pp. 747–752. Melbourne (USA) (2003)

  25. Raidl, G.R., Julstrom, B.A.: Edge sets: an effective evolutionary coding of spanning trees. IEEE Trans. Evol. Comput. 7(3), 225–239 (2003)

    Article  Google Scholar 

  26. Ramos, R.M., Alonso, S., Sicilia, J., Gonzalez, C.: The problem of the optimal biobjective spanning tree. Eur. J. Oper. Res. 111(3), 617–628 (1998)

    Article  Google Scholar 

  27. Requejo, C., Santos, E.: Greedy heuristics for the diameter-constrained minimum spanning tree problem. J. Math. Sci. 161, 930–943 (2009)

    Article  Google Scholar 

  28. Saha, S., Kumar, R.: Bounded-diameter MST instances with hybridization of multi-objective EA. Int. J. Comput. Appl. 18(4), 17–25 (2011)

    Google Scholar 

  29. Santos, A.C., Lucena, A., Ribeiro, C.C.: Solving diameter constrained minimum spanning tree problem in dense graphs. Lect. Notes Comput. Sci. 3059, 458–467 (2004)

    Article  Google Scholar 

  30. Sourd, F., Spanjaard, O.: A multiobjective branch-and-bound framework: application to the biobjective spanning tree problem. INFORMS J. Comput. 20(3), 472–484 (2008)

    Article  Google Scholar 

  31. Steiner, S., Radzik, T.: Computing all efficient solutions of the biobjective minimum spanning tree problem. Comput. Oper. Res. 35(1), 198–211 (2008)

    Article  Google Scholar 

  32. Thomas, B.W., Chen, H., Campbell, A.M.: Network design for time-constrained delivery. Nav. Res. Logist. 55, 493–515 (2008)

    Article  Google Scholar 

  33. Veldhuizen, D.A.V., Lamont, G.B.: On measuring multiobjective evolutionary algorithm performance. In: Proceedings of the Congress on Evolutionary Computation, vol. 1, pp. 204–211 (2000)

  34. Wismer, D., Haimes, Y., Ladson, L.: On a bicriterion formulation of the problems of integrated system identification and system optimization. IEEE Trans. Syst. Man Cybern. 1(3), 296–297 (1971)

    Article  Google Scholar 

  35. Zhou, G., Gen, M.: Genetic algorithm approach on multi-criteria minimum spanning tree problem. Eur. J. Oper. Res. 114, 141–152 (1999)

    Article  Google Scholar 

  36. Zhou, A., Qu, B.-Y., Li, H., Zhao, S.-Z., Suganthan, P.N., Zhang, Q.: Multiobjective evolutionary algorithms: a survey of the state of the art. Swarm Evol. Comput. 1(1), 32–49 (2011)

    Article  Google Scholar 

  37. Zitzler, E., Thiele, L., Laumanns, M., Fonseca, C., da Fonseca, V.: Performance assessment of multiobjective optimizers: an analysis and review. IEEE Trans. Evol. Comput. 7(2), 117132 (2003)

    Google Scholar 

  38. Zopounidis, C., Pardalos, P.M.: Handbook of Multicriteria Analysis, vol. 103. Springer, New York (2010)

    Book  Google Scholar 

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Authors and Affiliations

  1. ICD-LOSI, Université de Technologie de Troyes, 12, rue Marie Curie, CS 42060, 10004 , Troyes Cedex, France

    Andréa Cynthia Santos

  2. Universidade do Estado do Rio Grande do Norte, UERN, Rua Almino Afonso, 478, CEP 59.610-210 , Mossoró, RN, Brasil

    Diego Rocha Lima & Dario José Aloise

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  1. Andréa Cynthia Santos
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  2. Diego Rocha Lima
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  3. Dario José Aloise
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Correspondence to Andréa Cynthia Santos.

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Santos, A.C., Lima, D.R. & Aloise, D.J. Modeling and solving the bi-objective minimum diameter-cost spanning tree problem. J Glob Optim 60, 195–216 (2014). https://doi.org/10.1007/s10898-013-0124-4

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  • DOI: https://doi.org/10.1007/s10898-013-0124-4

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