Abstract
We study the sparse set conjecture for sets with low density. The sparse set conjecture states that P=NP if and only if there exists a sparse Turing hard set for NP. In this paper we study a weaker variant of the conjecture. We are interested in the consequences of NP having Turing hard sets of density f(n), for (unbounded) functions f(n), that are sub-polynomial, for example log(n). We establish a connection between Turing hard sets for NP with density f(n) and bounded nondeterminism: We prove that if NP has a Turing hard set of density f(n), then satisfiability is computable in polynomial time with O(log(n)*f(n c)) many nondeterministic bits for some constant c. As a consequence of the proof technique we obtain absolute results about the density of Turing hard sets for EXP. We show that no Turing hard set for EXP can have sub-polynomial density. On the other hand we show that these results are optimal w.r.t. relativizing computations. For unbounded functions f(n), there exists an oracle relative to which NP has a f(n) dense Turing hard tally set but still P≠NP.
Part of this research was done while visiting the Univ. Politècnica de Catalunya in Barcelona. Partially suported by the Dutch foundation for scientific research (NWO) through NFI Project ALADDIN, under contract number NF 62-376 and a TALENT stipend.
Part of this research was done while visiting the Univ. Politècnica de Catalunya in Barcelona.
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Buhrman, H., Hermo, M. (1995). On the sparse set conjecture for sets with low density. In: Mayr, E.W., Puech, C. (eds) STACS 95. STACS 1995. Lecture Notes in Computer Science, vol 900. Springer, Berlin, Heidelberg. https://doi.org/10.1007/3-540-59042-0_109
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DOI: https://doi.org/10.1007/3-540-59042-0_109
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