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The Complexity of Algorithmic Problems on Succinct Instances

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Computer Science

Abstract

Highly regular combinatorial objects can be represented advantageously by some kind of description shorter than their full standard encoding. For instance, graphs exhibiting enough regularities can be described using encodings substantially shorter than the full adjacency matrix. Anatural scheme for such succinct representations is by means of boolean circuits computing, as a boolean function, the values of individual bits of the binary encoding of the object. The complexity of many algorithmic problems changes drastically when this succinct representation is used to present the input. Two powerful lemmas quantifying exactly this increase of complexity are presented. These are applied to show that previous results in the area can be interpreted assufficient conditions for completeness in the logarithmic time and polynomial time counting hierarchies.

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This research was partially supported by the ESPRIT-II Basic Research Actions Program of the European Community under contract no. 3075 (project ALCOM).

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References

  1. J.L. Balcázar, J. Díaz, J. Gabarró: Structural Complexity I. EATCS Monographs on Theoretical Computer Science, vol. 11, Springer-Verlag (1988).

    Google Scholar 

  2. A. Chandra, D. Kozen, L. Stockmeyer: “Alternation”. Journal ACM 28 (1981), 114–133.

    Article  MathSciNet  MATH  Google Scholar 

  3. A. Chandra, L. Stockmeyer, U. Vishkin: “Constant depth reducibility”. SIAM Journal on Computing 13 (1984), 423–439.

    Article  MathSciNet  MATH  Google Scholar 

  4. B.S. Chlebus: “Domino-Tiling games”. Journal of Computer and System Sciences 32 (1986), 374–392.

    Article  MathSciNet  MATH  Google Scholar 

  5. M. Furst, J.B. Saxe, M. Sipser: “Parity, circuits, and the polynomial-time hierarchy”. Mathematical Systems Theory 17 (1984), 13–27.

    Article  MathSciNet  MATH  Google Scholar 

  6. H. Galperin, A. Wigderson: “Succinct representations of graphs”. Information and Control 56 (1983), 183–198.

    Article  MathSciNet  MATH  Google Scholar 

  7. M. Garey, D. Johnson: Computers and Intractability: A Guide to the Theory of NP-completeness. Freeman (1978).

    Google Scholar 

  8. J. Gill: “Computational complexity of probabilistic Turing machines”. SIAM Journal on Computing 6 (1977), 675–695.

    Article  MathSciNet  MATH  Google Scholar 

  9. E. Grädel: “Domino games with an application to the complexity of boolean algebras with bounded quantifier alternation”. Proc. 5th Symp. Theor. Aspects of Comp. Sci. LNCS 294, Springer-Verlag (1988), 98–107.

    Google Scholar 

  10. J. Hong: “On some deterministic space complexity problems”. SIAM Journal on Computing 11 (1982), 591–601.

    Article  MathSciNet  MATH  Google Scholar 

  11. N. Immerman: “Nondeterministic space is closed under complementation”. SIAM Journal on Computing 17 (1988), 935–938.

    Article  MathSciNet  MATH  Google Scholar 

  12. N. Jones, W. Laaser: “Complete problems for deterministic polynomial time”. Theoretical Computer Science 3 (1977), 105–117.

    Article  MathSciNet  MATH  Google Scholar 

  13. N.D. Jones, E. Lien, W.T. Laaser: “New problems complete for nondeterministic log space”. Mathematical Systems Theory 10 (1976), 1–17.

    Article  MathSciNet  MATH  Google Scholar 

  14. M. Kowaluk, K. Wagner: “Vector language: simple description of hard instances”. Proc. Math. Found, of Comp. Sci. LNCS 452, Springer-Verlag (1990), 378–384.

    Google Scholar 

  15. R.E. Ladner: “The circuit value problem is log space complete for P”. SIGACT News 7 (1975), 18–20.

    Article  Google Scholar 

  16. H.R. Lewis, Ch. Papadimitriou: “Symmetric space-bounded computation”. Theoretical Computer Science 19 (1982), 161–187.

    Article  MathSciNet  MATH  Google Scholar 

  17. A. Lozano: “iVP-hardness on succinct representations of graphs”. Bulletin of the EATCS 35 (1988), 158–163

    MATH  Google Scholar 

  18. A. Lozano, J.L. Balcázar: “The complexity of graph problems for succinctly represented graphs”. Proc. Graph-Theoretic Concepts in Comp. Sci. LNCS 411, Springer-Verlag (1989), 277–285.

    Article  Google Scholar 

  19. N. Lynch: “Logspace recognition and translation of parenthesis languages”. Journal A CM 24 (1977), 583–590.

    MathSciNet  MATH  Google Scholar 

  20. C.H. Papadimitriou, M. Yannakakis: “A note on succinct representations of graphs”. Information and Control 71 (1986), 181–185.

    Article  MathSciNet  MATH  Google Scholar 

  21. J.H. Reif: “Symmetric complementation”. Journal A CM 31 (1984), 401–421.

    MathSciNet  MATH  Google Scholar 

  22. M. Sipser: “Borel sets and circuit complexity”. Proc. 15th Symp. Theory of Comp. (1983), 61–69.

    Google Scholar 

  23. S. Skyum, L.G. Valiant: “A complexity theory based on boolean algebra”. Journal ACM 32 (1985), 484–502.

    Article  MathSciNet  MATH  Google Scholar 

  24. L.J. Stockmeyer: “The polynomial time hierarchy”. Theoretical Computer Science 3 (1977), 1–22.

    Article  MathSciNet  MATH  Google Scholar 

  25. R. Szelepcsényi: “The method of forced enumeration for nondeterministic automata”. Acta Informatica 26 (1988), 279–284.

    Article  MathSciNet  MATH  Google Scholar 

  26. J. Torán: “Succinct representations of counting problems”. 6th Int. Conference on Applied Algebra, Algebraic Algorithms, and Error Correcting Codes LNCS 357, Springer-Verlag (1988), 415–426.

    Article  Google Scholar 

  27. J. Torán: “Complexity classes defined by counting quantifiers”. Journal ACM 38, 753–774.

    Google Scholar 

  28. K. Wagner: “The complexity of problems concerning graphs with regularities”. Proc. Math. Found, of Comp. Sci. LNCS 176, Springer-Verlag (1984), 544–552.

    Google Scholar 

  29. K. Wagner: “The complexity of combinatorial problems with succinct input representation”. Acta Informatica 23 (1986), 325–356.

    Article  MathSciNet  MATH  Google Scholar 

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Authors and Affiliations

  1. Dep. Llenguatges i Sistemes Informàtics, Univ. Politècnica de Catalunya (ed. FIB), 08028, Barcelona, Spain

    José L. Balcázar, Antoni Lozano & Jacobo Torán

Authors
  1. José L. Balcázar
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  2. Antoni Lozano
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  3. Jacobo Torán
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Editor information

Editors and Affiliations

  1. University of Chile, Santiago, Chile

    Ricardo Baeza-Yates

  2. University of Arizona, Tucson, Arizona, USA

    Udi Manber

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© 1992 Springer Science+Business Media New York

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Balcázar, J.L., Lozano, A., Torán, J. (1992). The Complexity of Algorithmic Problems on Succinct Instances. In: Baeza-Yates, R., Manber, U. (eds) Computer Science. Springer, Boston, MA. https://doi.org/10.1007/978-1-4615-3422-8_30

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  • DOI: https://doi.org/10.1007/978-1-4615-3422-8_30

  • Publisher Name: Springer, Boston, MA

  • Print ISBN: 978-1-4613-6513-6

  • Online ISBN: 978-1-4615-3422-8

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