Circuit Lower Bounds, Help Functions, and the Remote Point Problem

TL;DR

This paper explores the limitations of algebraic and Boolean circuit models with help functions, linking lower bound proofs to the Remote Point Problem and establishing exponential lower bounds for certain algebraic models.

Contribution

It connects lower bound problems in algebraic and Boolean circuit models with help functions to the Remote Point Problem, providing new exponential lower bounds for specific algebraic models.

Findings

Lower bounds for ABPs with help polynomials are related to the Remote Point Problem in the rank metric.

Lower bounds for constant-depth circuits with help functions relate to the Remote Point Problem in the Hamming metric.

Exponential size lower bounds are shown for explicit polynomials in algebraic branching programs with help polynomials under degree restrictions.

Abstract

We investigate the power of Algebraic Branching Programs (ABPs) augmented with help polynomials, and constant-depth Boolean circuits augmented with help functions. We relate the problem of proving explicit lower bounds in both these models to the Remote Point Problem (introduced by Alon, Panigrahy, and Yekhanin (RANDOM '09)). More precisely, proving lower bounds for ABPs with help polynomials is related to the Remote Point Problem w.r.t. the rank metric, and for constant-depth circuits with help functions it is related to the Remote Point Problem w.r.t. the Hamming metric. For algebraic branching programs with help polynomials with some degree restrictions we show exponential size lower bounds for explicit polynomials.