Skip to main content
SpringerLink
Log in

Connectedness of Efficient Solutions in Multiple Objective Combinatorial Optimization

  • Published:
Journal of Optimization Theory and Applications Aims and scope Submit manuscript

Abstract

Connectedness of efficient solutions is a powerful property in multiple objective combinatorial optimization since it allows the construction of the complete efficient set using neighborhood search techniques. However, we show that many classical multiple objective combinatorial optimization problems do not possess the connectedness property in general, including, among others, knapsack problems (and even several special cases) and linear assignment problems. We also extend known non-connectedness results for several optimization problems on graphs like shortest path, spanning tree and minimum cost flow problems. Different concepts of connectedness are discussed in a formal setting, and numerical tests are performed for two variants of the knapsack problem to analyze the likelihood with which non-connected adjacency graphs occur in randomly generated instances.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

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

    MathSciNet  MATH  Google Scholar 

  2. Isermann, H.: The enumeration of the set of all efficient solutions for a linear multiple objective program. Oper. Res. Q. 28, 711–725 (1977)

    Article  MATH  Google Scholar 

  3. Naccache, P.: Connectedness of the set of nondominated outcomes. J. Optim. Theory Appl. 25, 459–467 (1978)

    Article  MathSciNet  MATH  Google Scholar 

  4. Helbig, S.: On the connectedness of the set of weakly efficient points of a vector optimization problem in locally convex spaces. J. Optim. Theory Appl. 65, 257–271 (1990)

    Article  MathSciNet  MATH  Google Scholar 

  5. Martins, E.Q.V.: On a multicriteria shortest path problem. Eur. J. Oper. Res. 16, 236–245 (1984)

    Article  MATH  Google Scholar 

  6. Ehrgott, M., Klamroth, K.: Connectedness of efficient solutions in multiple criteria combinatorial optimization. Eur. J. Oper. Res. 97, 159–166 (1997)

    Article  MATH  Google Scholar 

  7. Przybylski, A., Gandibleux, X., Ehrgott, M.: The biobjective integer minimum cost flow problem—incorrectness of Sedeño-Noda and González-Martín’s algorithm. Comput. Oper. Res. 33, 1459–1463 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  8. Sedeño-Noda, A., González-Martín, C.: An algorithm for the biobjective integer minimum cost flow problem. Comput. Oper. Res. 28, 139–156 (2001)

    Article  MathSciNet  MATH  Google Scholar 

  9. Pedersen, C.R.: Multicriteria discrete optimization—and related topics. Ph.D. Thesis, University of Aarhus, Denmark (2006)

  10. da Silva, C.G., Clímaco, J., Figueira, J.: Geometrical configuration of the Pareto frontier of bi-criteria {0,1}-knapsack problems. Working paper 16-2004, Institute for Systems and Computers Engineering, Coimbra, Portugal (2004)

  11. Gorski, J.: Multiple Objective Optimization and Implications for Single Objective Optimization. Shaker Verlag, Aachen (2010)

    MATH  Google Scholar 

  12. O’Sullivan, M., Walker, C.: Connecting efficient knapsacks—experiments with the equally-weighted bi-criteria knapsack problem. In: 39th Annual ORSNZ Conference, University of Auckland, Auckland, New Zealand, pp. 198–207 (2004)

    Google Scholar 

  13. Liefooghe, A., Paquete, L., Simões, M., Figueira, J.R.: Connectedness and local search for bicriteria knapsack problems. In: 11th European Conference on Evolutionary Computation in Combinatorial Optimisation. LNCS. Springer, Berlin (2011)

    Google Scholar 

  14. Paquete, L., Chiarandini, M., Stützle, T.: Pareto local optimum sets in the biobjective traveling salesman problem: An experimental study. In: Gandibleux, X., Sevaux, M., Srensen, K., T’kindt, V. (eds.) Metaheuristics for Multiobjective Optimisation. Lecture Notes in Economics and Mathematical Systems, vol. 535. Springer, Berlin (2004)

    Chapter  Google Scholar 

  15. Paquete, L., Stützle, T.: Clusters of nondominated solutions in multiobjective combinatorial optimization. In: 7th International Conference on Multi-Objective Programming and Goal Programming, Tours, France (2006)

    Google Scholar 

  16. Steuer, R.: Multiple Criteria Optimization. Theory, Computation, and Application. Wiley, New York (1985)

    Google Scholar 

  17. Ehrgott, M.: Multicriteria Optimization. Springer, Berlin (2005)

    MATH  Google Scholar 

  18. Gorski, J.: Analysis of the connectedness of Pareto-optimal solutions in multiple criteria combinatorial optimization. M.S. Thesis, University of Erlangen-Nuremberg, Germany (2004)

  19. Geist, D., Rodin, E.Y.: Adjacency of the 0-1 knapsack problem. Comput. Oper. Res. 19(8), 797–800 (1992)

    Article  MathSciNet  MATH  Google Scholar 

  20. Balinski, M.L., Russakoff, A.: On the assignment polytope. SIAM Rev. 16, 516–525 (1974)

    Article  MathSciNet  MATH  Google Scholar 

  21. Hausmann, D.: Adjacency on Polytopes in Combinatorial Optimization. Verlag Anton Hain, Königstein/Ts (1980)

    MATH  Google Scholar 

  22. Klamroth, K., Wiecek, M.: Dynamic programming approaches to the multiple criteria knapsack problem. Nav. Res. Logist. 47(1), 57–76 (2000)

    Article  MathSciNet  MATH  Google Scholar 

  23. Bazgan, C., Hugot, H., Vanderpooten, D.: Solving efficiently the 0-1 multi-objective knapsack problem. Comput. Oper. Res. 36(1), 260–279 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  24. Pedersen, C.R., Nielsen, L.R., Andersen, K.A.: On the bicriterion multi modal assignment problem. Working paper WP-2005-3, Department of Operations Research, University of Aarhus (2005)

Download references

Author information

Authors and Affiliations

  1. Arbeitsgruppe Optimierung und Approximation, Fachbereich C–Mathematik und Naturwissenschaften, Bergische Universität Wuppertal, Gaußstr. 20, 42119, Wuppertal, Germany

    Jochen Gorski & Kathrin Klamroth

  2. Department of Mathematics, University of Kaiserslautern, Postfach 3049, 67653, Kaiserslautern, Germany

    Stefan Ruzika

Authors
  1. Jochen Gorski
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Kathrin Klamroth
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Stefan Ruzika
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Kathrin Klamroth.

Additional information

Communicated by H. Benson.

The work of the three authors was supported by the project “Connectedness and Local Search for Multi-objective Combinatorial Optimization” founded by the Deutscher Akademischer Austausch Dienst and Conselho de Reitores das Universidades Portuguesas. In addition, the research of Stefan Ruzika was partially supported by Deutsche Forschungsgemeinschaft (DFG) grant HA 1737/7 “Algorithmik großer und komplexer Netzwerke”.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Gorski, J., Klamroth, K. & Ruzika, S. Connectedness of Efficient Solutions in Multiple Objective Combinatorial Optimization. J Optim Theory Appl 150, 475–497 (2011). https://doi.org/10.1007/s10957-011-9849-8

Download citation

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10957-011-9849-8

Keywords

Search

Navigation