Hardness as randomness: a survey of universal derandomization

TL;DR

This survey reviews recent progress in understanding whether probabilistic algorithms can be efficiently replaced by deterministic ones, highlighting the deep connections to circuit complexity and the challenges involved.

Contribution

It synthesizes recent developments in probabilistic complexity classes and discusses the equivalence between derandomization and circuit lower bounds.

Findings

Evidence supports P=BPP conjecture with low overhead

Derandomization is fundamentally linked to circuit lower bounds

Proving derandomization remains a major open challenge

Abstract

We survey recent developments in the study of probabilistic complexity classes. While the evidence seems to support the conjecture that probabilism can be deterministically simulated with relatively low overhead, i.e., that $P=BPP$, it also indicates that this may be a difficult question to resolve. In fact, proving that probabilistic algorithms have non-trivial deterministic simulations is basically equivalent to proving circuit lower bounds, either in the algebraic or Boolean models.