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Link to original content: https://doi.org/10.1007/s00224-022-10083-y
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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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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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