Structural Complexity of Multiobjective NP Search Problems

  • Krzysztof Fleszar
  • Christian Glaßer
  • Fabian Lipp
  • Christian Reitwießner
  • Maximilian Witek
Part of the Lecture Notes in Computer Science book series (LNCS, volume 7256)


An NP search problem is a multivalued function that maps instances to polynomially length-bounded solutions such that the validity of solutions is testable in polynomial time. NPMV g denotes the class of these functions.

There are at least two computational tasks associated with an NP search problem:

(i) Find out whether a solution exists.

(ii) Compute an arbitrary solution.

Further computational tasks arise in settings with multiple objectives, for example:

(iii) Compute a solution that is minimal w.r.t. the first objective,

while the second objective does not exceed some budget. Each such computational task defines a class of multivalued functions. We systematically investigate these classes and their relation to traditional complexity classes and classes of multivalued functions, like NP or max·P.

For multiobjective problems, some classes of computational tasks turn out to be equivalent to the function class NPMV g , some to the class of decision problems NP, and some to a seemingly new class that includes both NPMV g and NP. Under the assumption that certain exponential time classes are different, we show that there are computational tasks of multiobjective problems (more precisely functions in NPMV g ) that are Turing-inequivalent to any set.


Complexity Class Multivalued Function Partial Function Computational Task Multiobjective Optimization 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.


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  1. 1.
    Balcázar, J.L.: Self-reducibility structures and solutions of NP problems. Revista Matematica de la Universidad Complutense de Madrid 2(2-3), 175–184 (1989)MathSciNetzbMATHGoogle Scholar
  2. 2.
    Balcázar, J.L., Schöning, U.: Bi-immune sets for complexity classes. Mathematical Systems Theory 18(1), 1–10 (1985)MathSciNetzbMATHCrossRefGoogle Scholar
  3. 3.
    Beame, P., Cook, S.A., Edmonds, J., Impagliazzo, R., Pitassi, T.: The relative complexity of np search problems. Journal of Computer and System Sciences 57(1), 3–19 (1998)MathSciNetzbMATHCrossRefGoogle Scholar
  4. 4.
    Beigel, R., Bellare, M., Feigenbaum, J., Goldwasser, S.: Languages that are easier than their proofs. In: IEEE Symposium on Foundations of Computer Science, pp. 19–28 (1991)Google Scholar
  5. 5.
    Book, R.V., Long, T., Selman, A.L.: Quantitative relativizations of complexity classes. SIAM Journal on Computing 13, 461–487 (1984)MathSciNetzbMATHCrossRefGoogle Scholar
  6. 6.
    Borodin, A.B., Demers, A.J.: Some comments on functional self-reducibility and the NP hierarchy. Technical Report TR76-284, Cornell University, Department of Computer Science (1976)Google Scholar
  7. 7.
    Fenner, S., Green, F., Homer, S., Selman, A.L., Thierauf, T., Vollmer, H.: Complements of multivalued functions. Chicago Journal of Theor. Comp. Sc., Article 3 (1999)Google Scholar
  8. 8.
    Fenner, S., Homer, S., Ogihara, M., Selman, A.L.: Oracles that compute values. SIAM Journal on Computing 26, 1043–1065 (1997)MathSciNetzbMATHCrossRefGoogle Scholar
  9. 9.
    Glaßer, C., Reitwießner, C., Schmitz, H., Witek, M.: Approximability and Hardness in Multi-objective Optimization. In: Ferreira, F., Löwe, B., Mayordomo, E., Mendes Gomes, L. (eds.) CiE 2010. LNCS, vol. 6158, pp. 180–189. Springer, Heidelberg (2010)CrossRefGoogle Scholar
  10. 10.
    Hemaspaandra, L., Naik, A., Ogihara, M., Selman, A.L.: Computing solutions uniquely collapses the polynomial hierarchy. SIAM Journal on Computing 25, 697–708 (1996)MathSciNetzbMATHCrossRefGoogle Scholar
  11. 11.
    Hempel, H., Wechsung, G.: The operators min and max on the polynomial hierarchy. International Journal of Foundations of Computer Science 11(2), 315–342 (2000)MathSciNetCrossRefGoogle Scholar
  12. 12.
    Krentel, M.W.: The complexity of optimization problems. Journal of Computer and System Sciences 36, 490–509 (1988)MathSciNetzbMATHCrossRefGoogle Scholar
  13. 13.
    Papadimitriou, C.H., Yannakakis, M.: The complexity of restricted spanning tree problems. J. ACM 29(2), 285–309 (1982)MathSciNetzbMATHCrossRefGoogle Scholar
  14. 14.
    Selman, A.L.: A survey of one-way functions in complexity theory. Mathematical Systems Theory 25, 203–221 (1992)MathSciNetzbMATHCrossRefGoogle Scholar
  15. 15.
    Selman, A.L.: A taxonomy on complexity classes of functions. Journal of Computer and System Sciences 48, 357–381 (1994)MathSciNetzbMATHCrossRefGoogle Scholar
  16. 16.
    Selman, A.L.: Much ado about functions. In: Proceedings 11th Conference on Computational Complexity, pp. 198–212. IEEE Computer Society Press (1996)Google Scholar
  17. 17.
    Valiant, L.G.: Relative complexity of checking and evaluating. Information Processing Letters 5(1), 20–23 (1976)MathSciNetzbMATHCrossRefGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2012

Authors and Affiliations

  • Krzysztof Fleszar
    • 1
  • Christian Glaßer
    • 1
  • Fabian Lipp
    • 1
  • Christian Reitwießner
    • 1
  • Maximilian Witek
    • 1
  1. 1.Julius-Maximilians-Universität WürzburgGermany

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