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

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.

💡 Research Summary

The paper establishes a near‑quadratic lower bound on the size of syntactically multilinear arithmetic circuits that compute a particular explicit multilinear polynomial fₙ in n variables. Specifically, it proves that any such circuit must have size at least Ω(n² / log² n). This improves upon the previous best bound of Ω(n^{4/3} / log² n) by Raz, Shpilka, and Yehudayoff (RSY08) for the same family of hard polynomials.

The authors follow the overall strategy of RSY08: first they use a Baur‑Strassen style differentiation argument that, given a circuit Ψ of size s computing f, produces a syntactically multilinear circuit Ψ′ of size at most 5s with n output gates, each output computing the partial derivative f_i = ∂f/∂x_i. Because the circuit is syntactically multilinear, each variable x_i appears on a unique path to its corresponding output, which yields a clean separation of variables across the outputs.

The crucial technical step is to define, for each output i, a set U_i of “upper‑level” gates in Ψ′_i that depend on almost all variables (at least n – 6 log n) and have a child that depends on a moderate number of variables (between 6 log n and n – 6 log n). The set L_i of “lower‑level” gates consists of those children. A simple counting argument shows |U_i| ≥ |L_i|² because each gate has indegree at most two. RSY08 proved that each U_i must contain at least Ω(n^{1/3} / log n) gates, which yields the Ω(n^{4/3} / log² n) lower bound.

The present work strengthens this by proving a much larger lower bound on |U_i|. To achieve this, the authors translate the problem into a combinatorial question about families of subsets of