Nonuniform Reductions and NP-Completeness

Authors John M. Hitchcock, Hadi Shafei

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John M. Hitchcock
Hadi Shafei

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John M. Hitchcock and Hadi Shafei. Nonuniform Reductions and NP-Completeness. In 35th Symposium on Theoretical Aspects of Computer Science (STACS 2018). Leibniz International Proceedings in Informatics (LIPIcs), Volume 96, pp. 40:1-40:13, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2018)


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 NP0completeness, obtaining both separations and upper bounds for nonuniform completeness vs uniform complessness in NP. Under various hypotheses, we obtain the following separations: 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. 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 is stronger than a polynomial number of queries. 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 more powerful than a nonuniform reduction with fixed polynomial advice. 4. There is a set complete for NP under nonuniform many-one reductions with polynomial ad- vice, 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, the same statement for truth-table reductions and truth-table completeness also holds.
  • computational complexity
  • NP-completeness
  • reducibility
  • nonuniform complexity


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  1. L. Adleman. Two theorems on random polynomial time. In Proceedings of the 19th IEEE Symposium on Foundations of Computer Science, pages 75-83, 1978. Google Scholar
  2. M. Agrawal. Pseudo-random generators and structure of complete degrees. In Proceedings of the Seventeenth Annual IEEE Conference on Computational Complexity, pages 139-147. IEEE Computer Society, 2002. Google Scholar
  3. Manindra Agrawal and Osamu Watanabe. One-way functions and the berman-hartmanis conjecture. In Proceedings of the 24th Annual IEEE Conference on Computational Complexity, CCC 2009, Paris, France, 15-18 July 2009, pages 194-202. IEEE Computer Society, 2009. URL:
  4. E. Allender. The complexity of complexity. In Computability and Complexity - Essays Dedicated to Rodney G. Downey on the Occasion of His 60th Birthday, volume 10010 of Lecture Notes in Computer Science, pages 79-94, 2017. Google Scholar
  5. E. Allender, H. Buhrman, M. Koucký, D. van Melkebeek, and D. Ronneburger. Power from random strings. sicomp, 35:1467-1493, 2006. Google Scholar
  6. E. Allender and M. Strauss. Measure on small complexity classes with applications for BPP. In Proceedings of the 35th Symposium on Foundations of Computer Science, pages 807-818. IEEE Computer Society, 1994. Google Scholar
  7. K. Ambos-Spies and E. Mayordomo. Resource-bounded measure and randomness. In A. Sorbi, editor, Complexity, Logic and Recursion Theory, Lecture Notes in Pure and Applied Mathematics, pages 1-47. Marcel Dekker, New York, N.Y., 1997. Google Scholar
  8. K. Ambos-Spies, S. A. Terwijn, and X. Zheng. Resource bounded randomness and weakly complete problems. tcs, 172(1-2):195-207, 1997. Google Scholar
  9. L. Babai, L. Fortnow, N. Nisan, and A. Wigderson. BPP has subexponential simulations unless EXPTIME has publishable proofs. Computational Complexity, 3:307-318, 1993. Google Scholar
  10. J. L. Balcázar, J. Díaz, and J. Gabarró. Structural Complexity I. Springer-Verlag, Berlin, second edition, 1995. Google Scholar
  11. L. Berman and J. Hartmanis. On isomorphism and density of NP and other complete sets. SICOMP, 6(2):305-322, 1977. Google Scholar
  12. H. Buhrman and D. van Melkebeek. Hard sets are hard to find. jcss, 59(2):327-345, 1999. URL:
  13. Harry Buhrman, Benjamin J. Hescott, Steven Homer, and Leen Torenvliet. Non-uniform reductions. Theory Comput. Syst., 47(2):317-341, 2010. URL:
  14. S. Hirahara. Identifying an honest EXP^NP oracle among many. In Proceedings of the 30th Conference on Computational Complexity (CCC 2015), pages 244-263, 2015. Google Scholar
  15. J. M. Hitchcock. The size of SPP. tcs, 320(2-3):495-503, 2004. URL:
  16. J. M. Hitchcock and A. Pavan. Comparing reductions to NP-complete sets. Information and Computation, 205(5):694-706, 2007. URL:
  17. J. M. Hitchcock and A. Pavan. Hardness hypotheses, derandomization, and circuit complexity. Computational Complexity, 17(1):119-146, 2008. URL:
  18. J. M. Hitchcock and H. Shafei. Autoreducibility of NP-complete sets. In Proceedings of the 33rd International Symposium on Theoretical Aspects of Computer Science, pages 42:1-42:12. Leibniz International Proceedings in Informatics, 2016. Google Scholar
  19. D. W. Juedes and J. H. Lutz. Weak completeness in E and E₂. tcs, 143(1):149-158, 1995. Google Scholar
  20. V. Kabanets and J. Y. Cai. Circuit minimization problem. In Proceedings of the Thirty-Second Annual ACM Symposium on Theory of Computing, May 21-23, 2000, Portland, OR, USA, pages 73-79, 2000. Google Scholar
  21. R. M. Karp and R. J. Lipton. Turing machines that take advice. Enseign. Math., 28:191-201, 1982. Google Scholar
  22. A. Klivans and D. van Melkebeek. Graph nonisomorphism has subexponential size proofs unless the polynomial-time hierarchy collapses. SICOMP, 31(5):1501-1526, 2002. Google Scholar
  23. J. H. Lutz. Almost everywhere high nonuniform complexity. jcss, 44(2):220-258, 1992. Google Scholar
  24. J. H. Lutz. Observations on measure and lowness for Δ^P₂. Theory of Computing Systems, 30(4):429-442, 1997. Google Scholar
  25. J. H. Lutz. The quantitative structure of exponential time. In L. A. Hemaspaandra and A. L. Selman, editors, Complexity Theory Retrospective II, pages 225-254. Springer-Verlag, 1997. Google Scholar
  26. J. H. Lutz and E. Mayordomo. Cook versus Karp-Levin: Separating completeness notions if NP is not small. tcs, 164(1-2):141-163, 1996. Google Scholar
  27. C. H. Papadimitriou. Computational Complexity. Addison-Wesley, 1994. Google Scholar
  28. A. Pavan. Comparison of reductions and completeness notions. SIGACT News, 34(2):27-41, June 2003. URL:
  29. A. Pavan and A. L. Selman. Bi-immunity separates strong NP-completeness notions. Information and Computation, 188(1):116-126, 2004. Google Scholar
  30. D. Ronneberger. Kolmogorov Complexity and Derandomization. PhD thesis, Rutgers University, 2004. Google Scholar
  31. C. E. Shannon. The synthesis of two-terminal switching circuits. Bell System Technical Journal, 28(1):59-98, 1949. Google Scholar
  32. L. Valiant and V. Vazirani. NP is as easy as detecting unique solutions. tcs, 47(3):85-93, 1986. Google Scholar
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