Depth-4 Lower Bounds, Determinantal Complexity: A Unified Approach

Tavenas has recently proved that any nO(1)-variate and degree n polynomial in VP can be computed by a depth-4 ΣΠ[O( p n)]ΣΠ[ p n] circuit of size 2O( p n log n) [Tav13]. So to prove VP 6= VNP, it is sufficient to show that an explicit polynomial ∈ VNP of degree n requires 2ω( p n log n) size depth-4 circuits. Soon after Tavenas’s result, for two different explicit polynomials, depth-4 circuit size lower bounds of 2Ω( p n log n) have been proved ( [KSS13] and [FLMS13]). In particular, using combinatorial design Kayal et al. [KSS13] construct an explicit polynomial in VNP that requires depth-4 circuits of size 2Ω( p n log n) and Fournier et al. show that the iterated matrix multiplication polynomial (which is in VP) also requires 2Ω( p n log n) size depth-4 circuits. In this paper, we identify a simple combinatorial property such that any polynomial f that satisfies this property would achieve similar depth-4 circuit size lower bound. In particular, it does not matter whether f is in VP or in VNP. As a result, we get a simple unified lower bound analysis for the above mentioned polynomials. Another goal of this paper is to compare our current knowledge of the depth-4 circuit size lower bounds and the determinantal complexity lower bounds. Currently the best known determinantal complexity lower bound is Ω(n2) for Permanent of n × n matrix (which is a n2-variate and degree n polynomial) [CCL08]. We prove that the determinantal complexity of the iterated matrix multiplication polynomial is Ω(d n)where d is the number of matrices and n is the dimension of the matrices. So for d = n, we get that the iterated matrix multiplication polynomial achieves the current best known lower bounds in both fronts: depth-4 circuit size and determinantal complexity. Our result also settles the determinantal complexity of the iterated matrix multiplication polynomial to Θ(d n). To the best of our knowledge, a Θ(n) bound for the determinantal complexity for the iterated matrix multiplication polynomial was known only for a constant d > 1 [Jan11].