A Complexity Theory for Hard Enumeration Problems

TL;DR

This paper develops a foundational complexity theory for enumeration problems, introducing a hierarchy of classes and reduction notions to better analyze their computational difficulty.

Contribution

It proposes the first hierarchy of complexity classes and reduction concepts specifically for hard enumeration problems, filling a significant gap in the field.

Findings

Established a hierarchy of enumeration complexity classes.

Defined notions of reductions for enumeration problems.

Laid groundwork for future theoretical analysis.

Abstract

Complexity theory provides a wealth of complexity classes for analyzing the complexity of decision and counting problems. Despite the practical relevance of enumeration problems, the tools provided by complexity theory for this important class of problems are very limited. In particular, complexity classes analogous to the polynomial hierarchy and an appropriate notion of problem reduction are missing. In this work, we lay the foundations for a complexity theory of hard enumeration problems by proposing a hierarchy of complexity classes and by investigating notions of reductions for enumeration problems.