Lower Bounds for Monotone Arithmetic Circuits Via Communication Complexity.

Valiant (1980) showed that general arithmetic circuits with negation can be exponentially more powerful than monotone ones. We give the first improvement to this classical result: we construct a family of polynomials Pn in n variables, each of its monomials has non-negative coefficient, such that Pn can be computed by a polynomial-size depth-three formula but every monotone circuit computing it has size 2Ω(n1/4/log(n)). The polynomial Pn embeds the SINK∘ XOR function devised recently by Chattopadhyay, Mande and Sherif (2020) to refute the Log-Approximate-Rank Conjecture in communication complexity. To prove our lower bound for Pn, we develop a general connection between corruption of combinatorial rectangles by any function f ∘ XOR and corruption of product polynomials by a certain polynomial Pf that is an arithmetic embedding of f. This connection should be of independent interest. Using further ideas from communication complexity, we construct another family of set-multilinear polynomials fn,m such that both Fn,m − є· fn,m and Fn,m + є· fn,m have monotone circuit complexity 2Ω(n/log(n)) if є ≥ 2− Ω( m ) and Fn,m ∏i=1n (xi,1 +⋯+xi,m), with m = O( n/logn ). The polynomials fn,m have 0/1 coefficients and are in VNP. Proving such lower bounds for monotone circuits has been advocated recently by Hrubes (2020) as a first step towards proving lower bounds against general circuits via his new approach.