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Relating Partial and Complete Solutions and the Complexity of Computing Smallest Solutions

  • André Große
  • Jörg Rothe
  • Gerd Wechsung
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 2202)

Abstract

We prove that computing a single pair of vertices that are mapped onto each other by an isomorphism ϕ between two isomorphic graphs is as hard as computing ϕ itself.Th is result optimally improves upon a result of Gál et al.W e establish a similar, albeit slightly weaker, result about computing complete Hamiltonian cycles of a graph from partial Hamiltonian cycles.W e also show that computing the lexicographically first four-coloring for planar graphs is δ 2 p -hard. This result optimally improves upon a result of Khuller and Vazirani who prove this problem to be NP-hard, and conclude that it is not self-reducible in the sense of Schnorr, assuming P ≠ NP. W e discuss this application to non-self-reducibility and provide a general related result.

Keywords

partial solutions complexity of smallest solutions selfreducibility graph isomorphisms Hamiltonian cycles graph colorability 

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References

  1. 1.
    K. Appel and W. Haken. Every planar map is 4-colorable — 1: Discharging. Illinois J. Math, 21:429–490, 1977.zbMATHMathSciNetGoogle Scholar
  2. 2.
    K. Appel and W. Haken. Every planar map is 4-colorable — 2: Reducibility. Illinois J. Math, 21:491–567, 1977.zbMATHMathSciNetGoogle Scholar
  3. 3.
    A. Borodin and A. Demers. Some comments on functional self-reducibility and the NP hierarchy. Technical Report TR 76-284, Cornell Department of Computer Science, Ithaca, NY, July 1976.Google Scholar
  4. 4.
    A. Gál, S. Halevi, R. Lipton, and E. Petrank. Computing from partial solutions. In Proceedings of the 14th Annual IEEE Conference on Computational Complexity, pages 34–45. IEEE Computer Society Press, May 1999.Google Scholar
  5. 5.
    M. Garey and D. Johnson. Computers and Intractability: A Guide to the Theory of NP-Completeness. W. H. Freeman and Company, New York, 1979.zbMATHGoogle Scholar
  6. 6.
    M. Garey, D. Johnson, and L. Stockmeyer. Some simplified NP-complete graph problems. Theoretical Computer Science, 1:237–267, 1976.zbMATHCrossRefMathSciNetGoogle Scholar
  7. 7.
    A. Große. Partielle Lösungen NP-vollständiger Probleme. Diploma thesis, Friedrich-Schiller-Universität Jena, Institut für Informatik, Jena, Germany, December 1999. In German.Google Scholar
  8. 8.
    E. Hemaspaandra, A. Naik, M. Ogihara, and A. Selman. P-selective sets and reducing search to decision vs.self-reducibility. Journal of Computer and System Sciences, 53(2):194–209, 1996.zbMATHCrossRefMathSciNetGoogle Scholar
  9. 9.
    H. Hempel and G. Wechsung. The operators min and max on the polynomial hierarchy. International Journal of Foundations of Computer Science, 11(2):315–342, 2000.CrossRefMathSciNetGoogle Scholar
  10. 10.
    D. Joseph and P. Young. Self-reducibility: Effects of internal structure on computational complexity. In A. Selman, editor, Complexity Theory Retrospective, pages 82–107. Springer-Verlag, 1990.Google Scholar
  11. 11.
    S. Khuller and V. Vazirani. Planar graph coloring is not self-reducible, assuming P ≠ NP. Theoretical Computer Science, 88(1):183–189, 1991.zbMATHCrossRefMathSciNetGoogle Scholar
  12. 12.
    K. Ko. On self-reducibility and weak P-selectivity. Journal of Computer and System Sciences, 26(2):209–221, 1983.zbMATHCrossRefMathSciNetGoogle Scholar
  13. 13.
    M. Krentel. The complexity of optimization problems. Journal of Computer and System Sciences, 36:490–509, 1988.zbMATHCrossRefMathSciNetGoogle Scholar
  14. 14.
    A. Meyer and M. Paterson. With what frequency are apparently intractable problems difficult? Technical Report MIT/LCS/TM-126, MIT Laboratory for Computer Science, Cambridge, MA, 1979.Google Scholar
  15. 15.
    C. Papadimitriou. On the complexity of unique solutions. Journal of the ACM, 31(2):392–400, 1984.CrossRefMathSciNetGoogle Scholar
  16. 16.
    C. Schnorr. Optimal algorithms for self-reducible problems. In S. Michaelson and R. Milner, editors, Proceedings of the 3rd International Colloquium on Automata, Languages, and Programming, pages 322–337, University of Edinburgh, July 1976. Edinburgh University Press.Google Scholar
  17. 17.
    C. Schnorr. On self-transformable combinatorial problems, 1979. Presented at IEEE Symposium on Information Theory, Udine, and Symposium über Mathematische Optimierung, Oberwolfach.Google Scholar
  18. 18.
    A. Selman. Natural self-reducible sets. SIAM Journal on Computing, 17(5):989–996, 1988.CrossRefMathSciNetGoogle Scholar
  19. 19.
    L. Stockmeyer. Planar 3-colorability is NP-complete. SIGACT News, 5(3):19–25, 1973.CrossRefGoogle Scholar
  20. 20.
    H. Vollmer. On different reducibility notions for function classes. In Proceedings of the 11th Annual Symposium on Theoretical Aspects of Computer Science, pages 449–460. Springer-Verlag Lecture Notes in Computer Science #775, February 1994.Google Scholar
  21. 21.
    K. Wagner. More complicated questions about maxima and minima, and some closures of NP. Theoretical Computer Science, 51:53–80, 1987.zbMATHCrossRefMathSciNetGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2001

Authors and Affiliations

  • André Große
    • 1
  • Jörg Rothe
    • 2
  • Gerd Wechsung
    • 1
  1. 1.Institut für InformatikFriedrich-Schiller-Universität JenaJenaGermany
  2. 2.Math. InstitutHeinrich-Heine-Universität DüsseldorfDüsseldorfGermany

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