Lower Bounds of Algebraic Branching Programs and Layerization

TL;DR

This paper establishes a quadratic lower bound for unlayered algebraic branching programs and explores how layerization affects their size, revealing significant complexity differences between layered and unlayered models.

Contribution

It improves the lower bound for unlayered ABPs to 2(n^2) and demonstrates the impact of layerization on ABP size in both commutative and non-commutative settings.

Findings

Unlayered ABPs have 2(n^2) lower bound.

Layered ABPs can be exponentially larger than unlayered.

Layerization causes significant size increase in ABPs.

Abstract

In this paper we improve the lower bound of Chatterjee et al.\ (ECCC 2019) to an $\Omega(n^2)$ lower bound for unlayered Algebraic Branching Programs. We also study the impact layerization has on Algebraic Branching Programs. We exhibit a polynomial that has an unlayered ABP of size $O(n)$ but any layered ABP has size at least $\Omega(n\sqrt{n})$. We exhibit a similar dichotomy in the non-commutative setting where the unlayered ABP has size $O(n)$ and any layered ABP has size at least $\Omega(n\log n -\log^2 n)$.