Abstract
We investigate the question whether NE can be separated from the reduction closures of tally sets, sparse sets and NP. We show that (1) \(\mathrm{NE}\not\subseteq R^{\mathrm{NP}}_{n^{o(1)}-T}(\mathrm{TALLY})\); (2) \(\mathrm{NE}\not\subseteq R^{SN}_{m}(\mathrm{SPARSE})\); (3) \(\mathrm{NEXP}\not\subseteq \mathrm{P}^{\mathrm{NP}}_{n^{k}-T}/n^{k}\) for all k≥1; and (4) \(\mathrm{NE}\not\subseteq \mathrm{P}_{btt}(\mathrm{NP}\oplus\mathrm{SPARSE})\). Result (3) extends a previous result by Mocas to nonuniform reductions. We also investigate how different an NE-hard set is from an NP-set. We show that for any NP subset A of a many-one-hard set H for NE, there exists another NP subset A′ of H such that A′⊇ A and A′−A is not of sub-exponential density.
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A preliminary version of this work appeared as “Separating NE from Some Nonuniform Nondeterministic Complexity Classes” in the Proceedings of the 15th International Computing and Combinatorics Conference (COCOON’2009), Lecture Notes in Computer Science, vol. 5609, pp. 486–495, 2009.
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Fu, B., Li, A. & Zhang, L. Separating NE from some nonuniform nondeterministic complexity classes. J Comb Optim 22, 482–493 (2011). https://doi.org/10.1007/s10878-010-9327-5
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DOI: https://doi.org/10.1007/s10878-010-9327-5