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We investigate the question whether NE can be separated from the reduction closures of tally sets, sparse sets and NP. We show that (1) NE 6 ⊆ RNP no(1) −T (TALLY); (2)NE 6 ⊆ RSN m (SPARSE); and (3) NE 6 ⊆ PNP nk −T /nk for all k ≥ 1. 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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Journal of Combinatorial Optimization





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