Average-Case Complexity

TL;DR

This survey explores the average-case complexity of NP problems, discussing notions of good-on-average algorithms, completeness results, open questions, and the relation between worst-case and average-case complexities.

Contribution

It provides a comprehensive review of average-case complexity, including key results, open problems, and the relation between different degrees of average-case hardness in NP.

Findings

Completeness results link average-case ease of specific problems to all NP problems.

Open questions remain about basing average-case hardness on P≠NP assumptions.

Progress has been made in understanding the relation between various degrees of average-case complexity.

Abstract

We survey the average-case complexity of problems in NP. We discuss various notions of good-on-average algorithms, and present completeness results due to Impagliazzo and Levin. Such completeness results establish the fact that if a certain specific (but somewhat artificial) NP problem is easy-on-average with respect to the uniform distribution, then all problems in NP are easy-on-average with respect to all samplable distributions. Applying the theory to natural distributional problems remain an outstanding open question. We review some natural distributional problems whose average-case complexity is of particular interest and that do not yet fit into this theory. A major open question whether the existence of hard-on-average problems in NP can be based on the P$\neq$NP assumption or on related worst-case assumptions. We review negative results showing that certain proof techniques cannot prove such a result. While the relation between worst-case and average-case complexity for general NP problems remains open, there has been progress in understanding the relation between different "degrees" of average-case complexity. We discuss some of these "hardness amplification" results.