A Near-Optimal Depth-Hierarchy Theorem for Small-Depth Multilinear Circuits

TL;DR

This paper establishes tight size lower bounds for converting small-depth multilinear algebraic circuits, demonstrating exponential separation between formulas and circuits for certain polynomial computations, advancing understanding of algebraic circuit complexity.

Contribution

It proves near-optimal size blow-up bounds for depth reduction in multilinear circuits, extending previous results to a broader range of depths and strengthening known lower bounds.

Findings

Explicit polynomial examples require exponential size at lower depth

Depth reduction incurs exponential size blow-up in multilinear circuits

Results are tight up to constant factors in the exponent

Abstract

We study the size blow-up that is necessary to convert an algebraic circuit of product-depth $\Delta+1$ to one of product-depth $\Delta$ in the multilinear setting. We show that for every positive $\Delta = \Delta(n) = o(\log n/\log \log n),$ there is an explicit multilinear polynomial $P^{(\Delta)}$ on $n$ variables that can be computed by a multilinear formula of product-depth $\Delta+1$ and size $O(n)$, but not by any multilinear circuit of product-depth $\Delta$ and size less than $\exp(n^{\Omega(1/\Delta)})$. This result is tight up to the constant implicit in the double exponent for all $\Delta = o(\log n/\log \log n).$ This strengthens a result of Raz and Yehudayoff (Computational Complexity 2009) who prove a quasipolynomial separation for constant-depth multilinear circuits, and a result of Kayal, Nair and Saha (STACS 2016) who give an exponential separation in the case $\Delta = 1.$ Our separating examples may be viewed as algebraic analogues of variants of the Graph Reachability problem studied by Chen, Oliveira, Servedio and Tan (STOC 2016), who used them to prove lower bounds for constant-depth Boolean circuits.