Polynomial-time kernel reductions

TL;DR

This paper explores kernel reductions in computational complexity, analyzing their limitations, conditions for completeness, and their relation to other reduction types, highlighting their potential and constraints in problem equivalence.

Contribution

It provides a detailed analysis of kernel reductions, establishing their limitations, conditions for completeness, and their relationship with other reduction types in complexity theory.

Findings

Kernel reductions are weaker than many-one reductions.

Existence of complete problems depends on the number and size of equivalence classes.

Unconditional existence of complete problems under kernel reductions remains open.

Abstract

In the framework of computational complexity and in an effort to define a more natural reduction for problems of equivalence, we investigate the recently introduced kernel reduction, a reduction that operates on each element of a pair independently. This paper details the limitations and uses of kernel reductions. We show that kernel reductions are weaker than many-one reductions and provide conditions under which complete problems exist. Ultimately, the number and size of equivalence classes can dictate the existence of a kernel reduction. We leave unsolved the unconditional existence of a complete problem under polynomial-time kernel reductions for the standard complexity classes.