Symmetric Exponential Time Requires Near-Maximum Circuit Size: Simplified, Truly Uniform

TL;DR

This paper proves that symmetric exponential time classes require near-maximum circuit size by developing a simplified, uniform algorithm for the Range Avoidance Problem, leading to significant circuit lower bounds and pseudodeterministic constructions.

Contribution

It introduces a simple, uniform $ extsf{FS}_2P$ algorithm for the Range Avoidance Problem that works for all input sizes, establishing new circuit lower bounds and pseudodeterministic results.

Findings

Proves $ extsf{S}_2E ot ot i extsf{SIZE}[2^n/n]$ for all input sizes.

Establishes almost-everywhere near-maximum circuit lower bounds for certain complexity classes.

Provides pseudodeterministic constructions for various combinatorial objects.

Abstract

In a recent breakthrough, Chen, Hirahara and Ren prove that $\mathsf{S_2E}/_1 \not\subset \mathsf{SIZE}[2^n/n]$ by giving a single-valued $\mathsf{FS_2P}$ algorithm for the Range Avoidance Problem ($\mathsf{Avoid}$) that works for infinitely many input size $n$. Building on their work, we present a simple single-valued $\mathsf{FS_2P}$ algorithm for $\mathsf{Avoid}$ that works for all input size $n$. As a result, we obtain the circuit lower bound $\mathsf{S_2E} \not\subset {i.o.}$-$\mathsf{SIZE}[2^n/n]$ and many other corollaries: 1. Almost-everywhere near-maximum circuit lower bound for $\mathsf{\Sigma_2E} \cap \mathsf{\Pi_2E}$ and $\mathsf{ZPE}^{\mathsf{NP}}$. 2. Pseudodeterministic $\mathsf{FZPP}^{\mathsf{NP}}$ constructions for: Ramsey graphs, rigid matrices, pseudorandom generators, two-source extractors, linear codes, hard truth tables, and $K^{poly}$-random strings.