Abstract
In this paper we study the consequences of the existence of sparse hard sets for NP and other complexity classes under certain types of deterministic, randomized, and nondeterministic reductions. We show that if an NP-complete set is bounded truth-table reducible to some set that conjunctively reduces to a sparse set then P=NP. We next show that if an NP-complete set is bounded truth-table reducible to some set that randomly reduces (via a co-rp reduction) to some set that conjunctively reduces to a sparse set then RP=NP. Finally, we prove that if a coNP-complete set reduces via a nondeterministic polynomial time many-one reduction to a co-sparse set then PH=Θ p2 . On the other hand, we show that nondeterministic polynomial time many-one reductions to sparse sets are as powerful as nondeterministic Turing reductions to sparse sets.
Work done while visiting Universität Ulm. Supported in part by an Alexander von Humboldt research fellowship.
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Arvind, V., Köbler, J., Mundhenk, M. (1992). On bounded truth-table, conjunctive, and randomized reductions to sparse sets. In: Shyamasundar, R. (eds) Foundations of Software Technology and Theoretical Computer Science. FSTTCS 1992. Lecture Notes in Computer Science, vol 652. Springer, Berlin, Heidelberg. https://doi.org/10.1007/3-540-56287-7_101
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DOI: https://doi.org/10.1007/3-540-56287-7_101
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