On The Complexity of Enumeration

TL;DR

This paper explores the complexity classes of enumeration algorithms, demonstrating a hierarchy within IncP, equating incremental and polynomial delay under certain conditions, and linking uniform solution generation to probabilistic algorithms.

Contribution

It establishes a hierarchy within enumeration complexity classes and shows conditions under which incremental delay equals polynomial delay, advancing understanding of enumeration problem complexities.

Findings

IncP_1 equals DelayP with polynomial space under certain conditions

A strict hierarchy inside IncP is demonstrated using the Exponential Time Hypothesis

Uniform generation relates to probabilistic enumeration algorithms with polynomial delay and space

Abstract

We investigate the relationship between several enumeration complexity classes and focus in particular on problems having enumeration algorithms with incremental and polynomial delay (IncP and DelayP respectively). We show that, for some algorithms, we can turn an average delay into a worst case delay without increasing the space complexity, suggesting that IncP_1 = DelayP even with polynomially bounded space. We use the Exponential Time Hypothesis to exhibit a strict hierarchy inside IncP which gives the first separation of DelayP and IncP. Finally we relate the uniform generation of solutions to probabilistic enumeration algorithms with polynomial delay and polynomial space.