The Hardest Explicit Construction

TL;DR

This paper explores the complexity of explicit construction problems, establishing the class APEPP as central, and shows that derandomizing probabilistic existence proofs relates to constructing hard truth tables and circuit complexity.

Contribution

It identifies APEPP as the natural class for explicit constructions, connects high circuit complexity problems to APEPP completeness, and links derandomization to explicit truth table construction.

Findings

APEPP is the natural complexity class for explicit constructions.

Constructing high circuit complexity truth tables is APEPP-complete.

Derandomizing probabilistic proofs implies explicit construction of hard truth tables.

Abstract

We investigate the complexity of explicit construction problems, where the goal is to produce a particular object of size $n$ possessing some pseudorandom property in time polynomial in $n$. We give overwhelming evidence that $\bf{APEPP}$, defined originally by Kleinberg et al., is the natural complexity class associated with explicit constructions of objects whose existence follows from the probabilistic method, by placing a variety of such construction problems in this class. We then demonstrate that a result of Je\v{r}\'{a}bek on provability in Bounded Arithmetic, when reinterpreted as a reduction between search problems, shows that constructing a truth table of high circuit complexity is complete for $\bf{APEPP}$ under $\bf{P}^{\bf{NP}}$ reductions. This illustrates that Shannon's classical proof of the existence of hard boolean functions is in fact a $\textit{universal}$ probabilistic existence argument: derandomizing his proof implies a generic derandomization of the probabilistic method. As a corollary, we prove that $\bf{EXP}^{\bf{NP}}$ contains a language of circuit complexity $2^{n^{\Omega(1)}}$ if and only if it contains a language of circuit complexity $\frac{2^n}{2n}$. Finally, for several of the problems shown to lie in $\bf{APEPP}$, we demonstrate direct polynomial time reductions to the explicit construction of hard truth tables.