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The Multiobjective Shortest Path Problem Is NP-Hard, or Is It?

Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 10173)

Abstract

To show that multiobjective optimization problems like the multiobjective shortest path or assignment problems are hard, we often use the theory of \(\mathbf {NP} \)-hardness. In this paper we rigorously investigate the complexity status of some well-known multiobjective optimization problems and ask the question if these problems really are \(\mathbf {NP} \)-hard. It turns out, that most of them do not seem to be and for one we prove that if it is \(\mathbf {NP} \)-hard then this would imply \(\mathbf {P} = \mathbf {NP} \) under assumptions from the literature. We also reason why \(\mathbf {NP} \)-hardness might not be well suited for investigating the complexity status of intractable multiobjective optimization problems.

Keywords

Multiobjective Optimization Knapsack Problem Multiobjective Optimization Problem Single Objective Problem Travel Salesperson Problem 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

Notes

Acknowledgements

The author especially thanks Paolo Serafini for a very kind and indepth discussion of this topic. Also many thanks to Kathrin Klamroth and Michael Stiglmayr for many fruitful discussions on the complexity of multiobjective combinatorial optimization problems.

References

  1. 1.
    Arora, S., Barak, B.: Computational Complexity - A Modern Approach. Cambridge University Press, Cambridge (2009)CrossRefzbMATHGoogle Scholar
  2. 2.
    Beier, R., Röglin, H., Vöcking, B.: The smoothed number of Pareto optimal solutions in bicriteria integer optimization. In: Fischetti, M., Williamson, D.P. (eds.) IPCO 2007. LNCS, vol. 4513, pp. 53–67. Springer, Heidelberg (2007). doi: 10.1007/978-3-540-72792-7_5 CrossRefGoogle Scholar
  3. 3.
    Bökler, F., Ehrgott, M., Morris, C., Mutzel, P.: Output-sensitive complexity for multiobjective combinatorial optimization. J. Multi-Criteria Decis. Anal. (2017, accepted)Google Scholar
  4. 4.
    Bökler, F., Mutzel, P.: Output-sensitive algorithms for enumerating the extreme nondominated points of multiobjective combinatorial optimization problems. In: Bansal, N., Finocchi, I. (eds.) ESA 2015. LNCS, vol. 9294, pp. 288–299. Springer, Heidelberg (2015). doi: 10.1007/978-3-662-48350-3_25 CrossRefGoogle Scholar
  5. 5.
    Brunsch, T., Röglin, H.: Improved smoothed analysis of multiobjective optimization. In: ACM SToC. pp. 407–426 (2012)Google Scholar
  6. 6.
    Galand, L., Ismaili, A., Perny, P., Spanjaard, O.: Bidirectional preference-based search for multiobjective state space graph problems. In: SoCS 2013 (2013)Google Scholar
  7. 7.
    Garey, M.R., Johnson, D.S.: Computers and Intractability; A Guide to the Theory of NP-Completeness. Series of Books in the Mathematical Sciences, 1st edn. W. H. Freeman & Co., New York (1979)zbMATHGoogle Scholar
  8. 8.
    Guerriero, F., Musmanno, R.: Label correcting methods to solve multicriteria shortest path problems. J. Optim. Theory Appl. 111(3), 589–613 (2001)MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    Hansen, P.: Bicriterion path problems. In: Fandel, G., Gal, T. (eds.) Multiple Criteria Decision Making Theory and Application. Lecture Notes in Economics and Mathematical Systems, vol. 177, pp. 109–127. Springer, New York (1979)CrossRefGoogle Scholar
  10. 10.
    Horoba, C.: Exploring the runtime of an evolutionary algorithm for the multi-objective shortest path problem. Evol. Comput. 18(3), 357–381 (2010)CrossRefGoogle Scholar
  11. 11.
    Martins, E.Q.V.: On a multicriteria shortest path problem. Eur. J. Oper. Res. 16, 236–245 (1984)MathSciNetCrossRefzbMATHGoogle Scholar
  12. 12.
    Mohamed, C., Bassem, J., Taicir, L.: A genetic algorithm to solve the bicriteria shortest path problem. Electr. Notes Discret. Math. 36, 851–858 (2010)CrossRefzbMATHGoogle Scholar
  13. 13.
    Müller-Hannemann, M., Schulz, F., Wagner, D., Zaroliagis, C.: Timetable information: models and algorithms. In: Geraets, F., Kroon, F., Schoebel, A., Wagner, D., Zaroliagis, C.D. (eds.) Algorithmic Methods for Railway Optimization. LNCS, vol. 4359. Springer, Heidelberg (2007). doi: 10.1007/978-3-540-74247-0_3 Google Scholar
  14. 14.
    Nemhauser, G.L., Ullmann, Z.: Discrete dynamic programming and capital allocation. Manag. Sci. 15(9), 494–505 (1969)MathSciNetCrossRefzbMATHGoogle Scholar
  15. 15.
    Raith, A., Ehrgott, M.: A comparison of solution strategies for biobjective shortest path problems. Comput. Oper. Res. 36(4), 1299–1331 (2009)MathSciNetCrossRefzbMATHGoogle Scholar
  16. 16.
    Sanders, P., Mandow, L.: Parallel label-setting multi-objective shortest path search. In: Parallel & Distributed Processing (IPDPS), pp. 215–224. IEEE (2013)Google Scholar
  17. 17.
    Serafini, P.: Some considerations about computational complexity for multi-objective combinatorial problems. In: Jahn, J., Krabs, W. (eds.) Recent Advances and Historical Development of Vector Optimization. Lecture Notes in Economics and Mathematical Systems, vol. 294, pp. 222–231. Springer, Heidelberg (1986). doi: 10.1007/978-3-642-46618-2_15 CrossRefGoogle Scholar
  18. 18.
    Shekelyan, M., Jossé, G., Schubert, M.: Paretoprep: fast computation of path skyline queries. In: Advances in Spatial and Temporal Databases (2015)Google Scholar
  19. 19.
    Shekelyan, M., Jossé, G., Schubert, M., Kriegel, H.-P.: Linear Path Skyline Computation in Bicriteria Networks. In: Bhowmick, S.S., Dyreson, C.E., Jensen, C.S., Lee, M.L., Muliantara, A., Thalheim, B. (eds.) DASFAA 2014. LNCS, vol. 8421, pp. 173–187. Springer, Heidelberg (2014). doi: 10.1007/978-3-319-05810-8_12 CrossRefGoogle Scholar
  20. 20.
    Skriver, A.J.V.: A classification of bicriterion shortest path algorithms. Asia-Pac. J. Oper. Res. 17, 199–212 (2000)MathSciNetzbMATHGoogle Scholar
  21. 21.
    Tarapata, Z.: Selected multicriteria shortest path problems: An analysis of complexity, models and adaptation of standard algorithms. Int. J. Appl. Math. Comput. Sci. 17(2), 269–287 (2007)MathSciNetCrossRefzbMATHGoogle Scholar
  22. 22.
    Tsaggouris, G., Zaroliagis, C.D.: Multiobjective optimization: Improved FPTAS for shortest paths and non-linear objectives with applications. Algorithms and Computation 4288, 389–398 (2006)MathSciNetzbMATHGoogle Scholar
  23. 23.
    Warburton, A.: Approximation of Pareto optima in multiple-objective shortest-path problems. Oper. Res. 35(1), 70–79 (1987)MathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© Springer International Publishing AG 2017

Authors and Affiliations

  1. 1.Department of Computer ScienceTU DortmundDortmundGermany

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