On bounded truth-table, conjunctive, and randomized reductions to sparse sets

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 coNPcomplete set reduces via a nondeterministic polynomial time many-one reduction to a co-sparse set then PH = Θp2. 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.