In this paper we study the consequences of the existence of sparse hard sets for different 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 a set that conjunctively reduces to a sparse set then $\p=\np$. Relatedly, we show that if an $\np$-complete set is bounded-truth-table reducible to a set that co-rp reduces to some set that conjunctively reduces to a sparse set then $\rp=\np$. We also prove similar results under the (apparently) weaker assumption that some solution of the promise problem $(\ONESAT,\SAT)$ reduces via the mentioned reductions to a sparse set. Finally we consider nondeterministic polynomial time many-one reductions to sparse and co-sparse sets. We prove that if a $\coNP$-complete set reduces via a nondeterministic polynomial time many-one reduction to a co-sparse set then $\PH =\Theta^p_2$. 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.
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