Collapsing and Separating Completeness Notions under Average-Case and Worst-Case Hypotheses

Abstract

This paper presents the following results on sets that are complete for $\NP$.

\begin{enumerate}
\item If there is a problem in $\NP$ that requires $\twonO$ time at almost all lengths, then every many-one NP-complete set is complete under length-increasing reductions that are computed by polynomial-size circuits.
\item If there is a problem in $\CoNP$ that cannot be solved by polynomial-size nondeterministic circuits, then every many-one complete set is complete under length-increasing reductions that are computed by polynomial-size circuits.
\item If there exist a one-way permutation that is secure against subexponential-size circuits and there is a hard tally language in  $\NP \cap \CoNP$, then there is a Turing complete language for $\NP$
that is not many-one complete.
\end{enumerate}

Our first two results use worst-case hardness hypotheses whereas
earlier work that showed similar results relied on average-case or
almost-everywhere hardness assumptions.  The use of average-case and
worst-case hypotheses in the last result is unique as previous results
obtaining the same consequence relied on almost-everywhere hardness
results.