Autoreducibility of NP-Complete Sets

TL;DR

This paper investigates the polynomial-time autoreducibility properties of NP-complete sets under strong hypotheses, revealing separations between different autoreducibility notions and NP-completeness concepts.

Contribution

It establishes new separations between autoreducibility notions for NP-complete sets assuming the existence of p-generic sets in NP and coNP.

Findings

Existence of k-T-complete sets that are k-T autoreducible but not (k-tt) autoreducible.

Existence of k-tt-complete sets that are k-tt autoreducible but not (k-1)-tt or (k-2)-T autoreducible.

Existence of tt-complete sets that are tt-autoreducible but not btt-autoreducible.

Abstract

We study the polynomial-time autoreducibility of NP-complete sets and obtain separations under strong hypotheses for NP. Assuming there is a p-generic set in NP, we show the following: - For every $k \geq 2$, there is a $k$-T-complete set for NP that is $k$-T autoreducible, but is not $k$-tt autoreducible or $(k-1)$-T autoreducible. - For every $k \geq 3$, there is a $k$-tt-complete set for NP that is $k$-tt autoreducible, but is not $(k-1)$-tt autoreducible or $(k-2)$-T autoreducible. - There is a tt-complete set for NP that is tt-autoreducible, but is not btt-autoreducible. Under the stronger assumption that there is a p-generic set in NP $\cap$ coNP, we show: - For every $k \geq 2$, there is a $k$-tt-complete set for NP that is $k$-tt autoreducible, but is not $(k-1)$-T autoreducible. Our proofs are based on constructions from separating NP-completeness notions. For example, the construction of a 2-T-complete set for NP that is not 2-tt-complete also separates 2-T-autoreducibility from 2-tt-autoreducibility.