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.
![](http://media.springernature.com/m312/springer-static/image/art%3A10.1007%2Fs10898-013-0124-4/MediaObjects/10898_2013_124_Fig1_HTML.gif)
![](http://media.springernature.com/m312/springer-static/image/art%3A10.1007%2Fs10898-013-0124-4/MediaObjects/10898_2013_124_Fig2_HTML.gif)
![](http://media.springernature.com/m312/springer-static/image/art%3A10.1007%2Fs10898-013-0124-4/MediaObjects/10898_2013_124_Fig3_HTML.gif)
![](http://media.springernature.com/m312/springer-static/image/art%3A10.1007%2Fs10898-013-0124-4/MediaObjects/10898_2013_124_Fig4_HTML.gif)
![](http://media.springernature.com/m312/springer-static/image/art%3A10.1007%2Fs10898-013-0124-4/MediaObjects/10898_2013_124_Fig5_HTML.gif)
Similar content being viewed by others
References
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)
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)
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)
Chinchuluun, A., Pardalos, P.M.: A survey of recent developments in multiobjective optimization. Ann. Oper. Res. 154, 29–50 (2007)
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)
Cormen, T.H., Leiserson, C.E., Rivest, R., Stein, C.: Introduction to Algorithms, 3rd edn. The MIT Press, Cambridge (2009)
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)
Ehrgott, M., Gandibleux, X.: A survey and annotated bibliography of multiobjective combinatorial optimization. OR-Spektrum 22, 425–460 (2000)
Garey, M.R., Johnson, D.S.: Computers and Intractability: A Guide to the Theory of NP-Completeness. W.H. Freeman, New York (1979)
Gouveia, L., Magnanti, T.L.: Network flow models for designing diameter-constrained minimum-spanning and Steiner trees. Networks 41, 159–173 (2003)
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)
Handler, G.Y.: Minimax location of a facility in an undirected graph. Transp. Sci. 7, 287–293 (1978)
Hassin, R., Tamir, A.: On the minimum diameter spanning tree problem. Inf. Process. Lett. 53(2), 109–111 (1995)
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)
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)
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)
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)
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)
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)
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)
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)
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)
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)
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)
Raidl, G.R., Julstrom, B.A.: Edge sets: an effective evolutionary coding of spanning trees. IEEE Trans. Evol. Comput. 7(3), 225–239 (2003)
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)
Requejo, C., Santos, E.: Greedy heuristics for the diameter-constrained minimum spanning tree problem. J. Math. Sci. 161, 930–943 (2009)
Saha, S., Kumar, R.: Bounded-diameter MST instances with hybridization of multi-objective EA. Int. J. Comput. Appl. 18(4), 17–25 (2011)
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)
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)
Steiner, S., Radzik, T.: Computing all efficient solutions of the biobjective minimum spanning tree problem. Comput. Oper. Res. 35(1), 198–211 (2008)
Thomas, B.W., Chen, H., Campbell, A.M.: Network design for time-constrained delivery. Nav. Res. Logist. 55, 493–515 (2008)
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)
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)
Zhou, G., Gen, M.: Genetic algorithm approach on multi-criteria minimum spanning tree problem. Eur. J. Oper. Res. 114, 141–152 (1999)
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)
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)
Zopounidis, C., Pardalos, P.M.: Handbook of Multicriteria Analysis, vol. 103. Springer, New York (2010)
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
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
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s10898-013-0124-4