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The Complexity of Unions of Disjoint Sets

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Part of the book series: Lecture Notes in Computer Science ((LNTCS,volume 4393))

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Abstract

This paper is motivated by the open question whether the union of two disjoint NP-complete sets always is NP-complete. We discover that such unions retain much of the complexity of their single components. More precisely, they are complete with respect to more general reducibilities.

Moreover, we approach the main question in a more general way: We analyze the scope of the complexity of unions of m-equivalent disjoint sets. Under the hypothesis that NE ≠ coNE, we construct degrees in NP where our main question has a positive answer, i.e., these degrees are closed under unions of disjoint sets.

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References

  1. Agrawal, M.: Pseudo-random generators and structure of complete degrees. In: IEEE Conference on Computational Complexity, pp. 139–147. IEEE Computer Society Press, Los Alamitos (2002)

    Google Scholar 

  2. Ambos-Spies, K.: P-mitotic sets. In: Börger, E., Rödding, D., Hasenjaeger, G. (eds.) Rekursive Kombinatorik 1983. LNCS, vol. 171, pp. 1–23. Springer, Heidelberg (1984)

    Google Scholar 

  3. Blass, A., Gurevich, Y.: On the unique satisfiability problem. Information and Control 82, 80–88 (1982)

    Article  MathSciNet  Google Scholar 

  4. Berman, L., Hartmanis, J.: On isomorphism and density of NP and other complete sets. SIAM Journal on Computing 6, 305–322 (1977)

    Article  MathSciNet  MATH  Google Scholar 

  5. Buhrman, H., Hoene, A., Torenvliet, L.: Splittings, robustness, and structure of complete sets. SIAM Journal on Computing 27, 637–653 (1998)

    Article  MathSciNet  MATH  Google Scholar 

  6. Boneh, D., Venkatesan, R.: Rounding in lattices and its cryptographic applications. In: SODA, pp. 675–681 (1997)

    Google Scholar 

  7. Book, R.V., et al.: Inclusion complete tally languages and the hartmanis-berman conjecture. Mathematical Systems Theory 11, 1–8 (1977)

    Article  MathSciNet  MATH  Google Scholar 

  8. Cai, J.-Y., et al.: The boolean hierarchy I: Structural properties. SIAM Journal on Computing 17, 1232–1252 (1988)

    Article  MathSciNet  MATH  Google Scholar 

  9. Downey, R.G., Fortnow, L.: Uniformly hard languages. Theoretical Computer Science 298(2), 303–315 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  10. Fortnow, L., Rogers, J.: Separability and one-way functions. Computational Complexity 11(3-4), 137–157 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  11. Glaßer, C., et al.: Redundancy in complete sets. In: Durand, B., Thomas, W. (eds.) STACS 2006. LNCS, vol. 3884, pp. 444–454. Springer, Heidelberg (2006)

    Chapter  Google Scholar 

  12. Grollmann, J., Selman, A.L.: Complexity measures for public-key cryptosystems. SIAM Journal on Computing 17(2), 309–335 (1988)

    Article  MathSciNet  MATH  Google Scholar 

  13. Glaßer, C., Travers, S.: Machines that can output empty words. In: Královič, R., Urzyczyn, P. (eds.) MFCS 2006. LNCS, vol. 4162, pp. 436–446. Springer, Heidelberg (2006)

    Chapter  Google Scholar 

  14. Gundermann, T., Wechsung, G.: Nondeterministic Turing machines with modified acceptance. In: Wiedermann, J., Gruska, J., Rovan, B. (eds.) MFCS 1986. LNCS, vol. 233, pp. 396–404. Springer, Heidelberg (1986)

    Chapter  Google Scholar 

  15. Hitchcock, J., Pavan, A.: Comparing reductions to NP-complete sets. In: Bugliesi, M., et al. (eds.) ICALP 2006. LNCS, vol. 4051, pp. 465–476. Springer, Heidelberg (2006)

    Chapter  Google Scholar 

  16. Köbler, J., Schöning, U.: High sets for NP. In: Advances in Algorithms, Languages, and Complexity, pp. 139–156 (1997)

    Google Scholar 

  17. Köbler, J., Schöning, U., Wagner, K.W.: The difference and the truth-table hierarchies for NP. RAIRO Inform. Théor. 21, 419–435 (1987)

    MATH  Google Scholar 

  18. Ladner, R.E., Lynch, N.A., Selman, A.L.: A comparison of polynomial time reducibilities. Theoretical Computer Science 1, 103–123 (1975)

    Article  MathSciNet  MATH  Google Scholar 

  19. Long, T.J.: On some Polynomial Time Reducibilities. PhD thesis, Purdue University, Lafayette, Ind. (1978)

    Google Scholar 

  20. Schöning, U.: A low and a high hierarchy within NP. Journal of Computer and System Sciences 27(1), 14–28 (1983)

    Article  MathSciNet  MATH  Google Scholar 

  21. Selman, A.L.: P-selective sets, tally languages, and the behavior of polynomial-time reducibilities on NP. Mathematical Systems Theory 13, 55–65 (1979)

    Article  MathSciNet  MATH  Google Scholar 

  22. Selman, A.L.: Natural self-reducible sets. SIAM Journal on Computing 17(5), 989–996 (1988)

    Article  MathSciNet  Google Scholar 

  23. Valiant, L.G.: Relative complexity of checking and evaluation. Information Processing Letters 5, 20–23 (1976)

    Article  MathSciNet  MATH  Google Scholar 

  24. Wagner, K.W., Wechsung, G.: On the boolean closure of NP. In: Budach, L. (ed.) FCT 1985. LNCS, vol. 199, pp. 485–493. Springer, Heidelberg (1985)

    Google Scholar 

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Authors and Affiliations

  1. Universität Würzburg, Germany

    Christian Glaßer, Stephen Travers & Klaus W. Wagner

  2. University at Buffalo, USA

    Alan L. Selman

Authors
  1. Christian Glaßer
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  2. Alan L. Selman
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  3. Stephen Travers
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  4. Klaus W. Wagner
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Editor information

Wolfgang Thomas Pascal Weil

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Glaßer, C., Selman, A.L., Travers, S., Wagner, K.W. (2007). The Complexity of Unions of Disjoint Sets. In: Thomas, W., Weil, P. (eds) STACS 2007. STACS 2007. Lecture Notes in Computer Science, vol 4393. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-540-70918-3_22

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  • DOI: https://doi.org/10.1007/978-3-540-70918-3_22

  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-540-70917-6

  • Online ISBN: 978-3-540-70918-3

  • eBook Packages: Computer ScienceComputer Science (R0)

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