Unbalancing Sets and an Almost Quadratic Lower Bound for Syntactically Multilinear Arithmetic Circuits

TL;DR

This paper establishes a nearly quadratic lower bound on the size of syntactically multilinear arithmetic circuits computing explicit polynomials, advancing the understanding of circuit complexity with improved bounds over previous results.

Contribution

It introduces an asymptotically optimal lower bound for a generalized extremal set theory problem, leading to a stronger lower bound for multilinear circuit size.

Findings

Proves an n^2/log^2 n lower bound for explicit multilinear polynomials.

Improves previous lower bounds from n^{4/3}/log^2 n to nearly quadratic.

Connects circuit complexity lower bounds to extremal set theory results.

Abstract

We prove a lower bound of $\Omega(n^2/\log^2 n)$ on the size of any syntactically multilinear arithmetic circuit computing some explicit multilinear polynomial $f(x_1, \ldots, x_n)$. Our approach expands and improves upon a result of Raz, Shpilka and Yehudayoff ([RSY08]), who proved a lower bound of $\Omega(n^{4/3}/\log^2 n)$ for the same polynomial. Our improvement follows from an asymptotically optimal lower bound for a generalized version of Galvin's problem in extremal set theory.