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

TL;DR

This paper explores the relationships between different NP completeness notions under various average-case and worst-case hardness hypotheses, revealing conditions where certain completeness notions collapse or separate.

Contribution

It introduces new results linking average-case and worst-case hardness assumptions to the equivalence or separation of NP completeness notions, using novel hypotheses.

Findings

Under certain hardness assumptions, all NP-complete sets are complete under polynomial-size circuit reductions.

Hardness assumptions imply the existence of NP languages that are Turing complete but not many-one complete.

The results connect average-case and worst-case hypotheses to the structure of NP completeness notions.

Abstract

This paper presents the following results on sets that are complete for NP. 1. If there is a problem in NP that requires exponential 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. 2. 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. 3. If there exist a one-way permutation that is secure against subexponential-size circuits and there is a hard tally language in NP intersect coNP, then there is a Turing complete language for NP that is not many-one complete. 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.