Favorite Theorems: Multilinear Circuits

In the past decade we have seen a strong program in algebraic circuit complexity. If you just define circuits using multiplication and addition gates, sometimes you can use the algebraic structure to get lower bounds that prove much more difficult in the Boolean case. For October's favorite theorem we highlight a couple of Ran Raz's results.

Separation of Multilinear Circuit and Formula Size by Ran Raz
Multi-linear formulas for permanent and determinant are of super-polynomial size by Raz Raz

The main result of the first paper is basically in the title. Raz creates a polynomial function f that can be computed by a polynomial multilinear circuit of depth O(log² n) but cannot be computed by any multilinear formula. (A formula, unlike a circuit, cannot use the output of a gate more than once). His proof works by examining the rank of the partial derivative matrix of a function chosen in a specified random way.

Ran Raz followed up with Amir Shpilka and Amir Yehudayoff to show lower bounds for syntactically multilinear circuits and exponential bounds for constant-depth multilinear circuits.

The second paper showed the most well-known of the algebraic functions (permanent and determinant) do not have poly-size multilinear formulas. Raz again starts with the partial derivative matrix, this time combined with random restrictions.

More recent work in algebraic circuit complexity focuses on the VP versus VNP problem, the algebraic version of P v NP. VP = VNP if we can solve the permanent on poly-size algebraic circuits. Proving VP ≠ VNP is a goal of the Geometric Complexity Theory program as well as a consequence of tighter bounds on constant-depth algebraic circuits.