On the structure of linear-time reducibility

TL;DR

This paper extends Ladner's theorem to linear-time reductions, demonstrating that under P ≠ NP, there exist problems in linear-time classes that are neither in P nor NP-complete, similar to polynomial-time results.

Contribution

It generalizes Ladner's theorem to linear-time reductions on RAMs and Turing machines, enabling separation results in linear-time complexity classes.

Findings

Existence of intermediate problems in linear-time classes under P ≠ NP

Linear-time reductions can be used for separation results

Results analogous to Ladner's theorem for polynomial time

Abstract

In 1975, Ladner showed that under the hypothesis that P is not equal to NP, there exists a language which is neither in P, nor NP-complete. This result was latter generalized by Schoning and several authors to various polynomial-time complexity classes. We show here that such results also apply to linear-time reductions on RAMs (resp. Turing machines), and hence allow for separation results in linear-time classes similar to Ladner's ones for polynomial time.