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Nonuniform Reductions and NP-Completeness

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Abstract

Nonuniformity is a central concept in computational complexity with powerful connections to circuit complexity and randomness. Nonuniform reductions have been used to study the isomorphism conjecture for NP and completeness for larger complexity classes. We study the power of nonuniform reductions for NP-completeness, obtaining both separations and upper bounds for nonuniform completeness vs uniform complessness in NP. Under various hypotheses, we obtain the following separations:

  1. 1.

    There is a set complete for NP under nonuniform many-one reductions, but not under uniform many-one reductions. This is true even with a single bit of nonuniform advice.

  2. 2.

    There is a set complete for NP under nonuniform many-one reductions with polynomial-size advice, but not under uniform Turing reductions. That is, polynomial nonuniformity cannot be replaced by a polynomial number of queries.

  3. 3.

    For any fixed polynomial p(n), there is a set complete for NP under uniform 2-truth-table reductions, but not under nonuniform many-one reductions that use p(n) advice. That is, giving a uniform reduction a second query makes it impossible to simulate by a nonuniform reduction with fixed polynomial advice.

  4. 4.

    There is a set complete for NP under nonuniform many-one reductions with polynomial advice, but not under nonuniform many-one reductions with logarithmic advice. This hierarchy theorem also holds for other reducibilities, such as truth-table and Turing.

We also consider uniform upper bounds on nonuniform completeness. Hirahara (2015) showed that unconditionally every set that is complete for NP under nonuniform truth-table reductions that use logarithmic advice is also uniformly Turing-complete. We show that under a derandomization hypothesis, every set that is complete for NP under nonuniform truth-table reductions is also uniformly truth-table complete.

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References

  1. Adleman, L.: Two theorems on random polynomial time. In: Proceedings of the 19th IEEE Symposium on Foundations of Computer Science, pp 75–83 (1978)

  2. Agrawal, M.: Pseudo-random generators and structure of complete degrees. In: Proceedings of the Seventeenth Annual IEEE Conference on Computational Complexity, pp 139–147. IEEE Computer Society (2002)

  3. Agrawal, M., Watanabe, O.: One-way functions and the Berman-Hartmanis conjecture. In: Proceedings of the 24th Annual IEEE Conference on Computational Complexity, pp 194–202 (2009)

  4. Allender, E.: When worlds collide: Derandomization, lower bounds, and Kolmogorov complexity. In: Proceedings of the 21st Conference on Foundations of Software Technology and Theoretical Computer Science, pp 1–15. Springer-Verlag (2001)

  5. Allender, E.: The complexity of complexity. In: Computability and Complexity - Essays Dedicated to Rodney G. Downey on the Occasion of His 60th Birthday, Lecture Notes in Computer Science, vol. 10010, pp 79–94 (2017)

  6. Allender, E., Buhrman, H., Koucký, M., van Melkebeek, D., Ronneburger, D.: Power from random strings. SIAM Journal on Computing 35, 1467–1493 (2006)

    Article  MathSciNet  Google Scholar 

  7. Ambos-Spies, K., Mayordomo, E.: Resource-bounded measure and randomness. In: Sorbi, A. (ed.) Complexity, Logic and Recursion Theory, Lecture Notes in Pure and Applied Mathematics, pp 1–47. Marcel Dekker, New York, N.Y. (1997)

  8. Ambos-Spies, K., Terwijn, S.A., Zheng, X.: Resource bounded randomness and weakly complete problems. Theoretical Computer Science 172(1–2), 195–207 (1997)

    Article  MathSciNet  Google Scholar 

  9. Babai, L., Fortnow, L., Nisan, N., Wigderson, A.: BPP has subexponential simulations unless EXPTIME has publishable proofs. Comput. Complex. 3, 307–318 (1993)

    Article  MathSciNet  Google Scholar 

  10. Balcázar, J.L., Díaz, J., Gabarró, J.: Structural Complexity I, 2nd edn. Springer-Verlag, Berlin (1995)

    Book  Google Scholar 

  11. Berman, L.: On the structure of complete sets: Almost everywhere complexity and infinitely often speedup. In: Proceedings of the Seventeenth Annual Conference on Foundations of Computer Science, pp 76–80 (1976)

  12. Berman, L., Hartmanis, J.: On isomorphism and density of NP and other complete sets. SIAM Journal on Computing 6(2), 305–322 (1977)

    Article  MathSciNet  Google Scholar 

  13. Buhrman, H., Hescott, B., Homer, S., Torenvliet, L.: Non-uniform reductions. Theory Comput. Syst. 47(2), 317–341 (2010)

    Article  MathSciNet  Google Scholar 

  14. Buhrman, H., van Melkebeek, D.: Hard sets are hard to find. J. Comput. Syst. Sci. 59(2), 327–345 (1999)

    Article  MathSciNet  Google Scholar 

  15. Hirahara, S.: Identifying an honest EXPNP oracle among many. In: Proceedings of the 30th Conference on Computational Complexity (CCC 2015), pp 244–263 (2015)

  16. Hitchcock, J.M., Pavan, A.: Comparing reductions to NP-complete sets. Inf. Comput. 205(5), 694–706 (2007)

    Article  MathSciNet  Google Scholar 

  17. Hitchcock, J.M., Pavan, A.: Hardness hypotheses, derandomization, and circuit complexity. Comput. Complex. 17(1), 119–146 (2008)

    Article  MathSciNet  Google Scholar 

  18. Hitchcock, J.M., Shafei, H.: Autoreducibility of NP-complete sets under strong hypotheses. Comput. Complex. 27(1), 63–97 (2018)

    Article  MathSciNet  Google Scholar 

  19. Juedes, D.W., Lutz, J.H.: Weak completeness in E and E2. Theor. Comput. Sci. 143(1), 149–158 (1995)

    Article  Google Scholar 

  20. Kabanets, V., Cai, J.Y.: Circuit minimization problem. In: Proceedings of the Thirty-Second Annual ACM Symposium on Theory of Computing, May 21-23, 2000, Portland, OR, USA, pp 73–79 (2000)

  21. Karp, R.M., Lipton, R.J.: Turing machines that take advice. Enseign. Math. 28, 191–201 (1982)

    MathSciNet  MATH  Google Scholar 

  22. Klivans, A., van Melkebeek, D.: Graph nonisomorphism has subexponential size proofs unless the polynomial-time hierarchy collapses. SIAM J. Comput. 31(5), 1501–1526 (2002)

    Article  MathSciNet  Google Scholar 

  23. Lutz, J.H.: Almost everywhere high nonuniform complexity. J. Comput. Syst. Sci. 44(2), 220–258 (1992)

    Article  MathSciNet  Google Scholar 

  24. Lutz, J.H.: The quantitative structure of exponential time. In: Hemaspaandra, L.A., Selman, A.L. (eds.) Complexity Theory Retrospective II, pp 225–254. Springer-Verlag (1997)

  25. Lutz, J.H., Karp-Levin, E. Mayordomo. Cook versus: Separating completeness notions if NP is not small. Theor. Comput. Sci. 164(1–2), 141–163 (1996)

    Article  Google Scholar 

  26. Papadimitriou, C.H.: Comput. CompleȦddison-Wesley (1994)

  27. Pavan, A.: Comparison of reductions and completeness notions. SIGACT News 34(2), 27–41 (2003)

    Article  Google Scholar 

  28. Pavan, A., Selman, A.L.: Bi-immunity separates strong NP-completeness notions. Inform. Comput. 188(1), 116–126 (2004)

    Article  MathSciNet  Google Scholar 

  29. Valiant, L., Vazirani, V.: NP is as easy as detecting unique solutions. Theor. Comput. Sci. 47(3), 85–93 (1986)

    Article  MathSciNet  Google Scholar 

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Acknowledgements

We thank the anonymous referees for many helpful comments and suggestions.

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Authors and Affiliations

  1. Department of Computer Science, University of Wyoming, Laramie, WY, 82071, USA

    John M. Hitchcock

  2. Department of Mathematics and Computer Science, Northern Michigan University, Marquette, MI, 49855, USA

    Hadi Shafei

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  1. John M. Hitchcock
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  2. Hadi Shafei
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Correspondence to John M. Hitchcock.

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Hitchcock, J.M., Shafei, H. Nonuniform Reductions and NP-Completeness. Theory Comput Syst 66, 743–757 (2022). https://doi.org/10.1007/s00224-022-10083-y

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