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

Abstract

Tavenas (Proceedings of mathematical foundations of computer science (MFCS), 2013) has recently proved that any $n^{O(1)}$-variate and degree n polynomial in $\mathsf {VP}$ can be computed by a depth-4 $\Sigma \Pi \Sigma \Pi$ circuit of size $2^{O(\sqrt{n}\log n)}$. So, to prove $\mathsf {VP}\ne \mathsf {VNP}$ it is sufficient to show that an explicit polynomial in $\mathsf {VNP}$ of degree n requires $2^{\omega (\sqrt{n}\log n)}$ size depth-4 circuits. Soon after Tavenas’ result, for two different explicit polynomials, depth-4 circuit-size lower bounds of $2^{\Omega (\sqrt{n}\log n)}$ have been proved (see Kayal et al. in Proceedings of symposium on theory of computing, ACM, 2014b. http://doi.acm.org/10.1145/2591796.2591847; Fournier et al. in Proceedings of symposium on theory of computing, ACM, 2014). In particular, using a combinatorial design Kayal et al. (2014b) construct an explicit polynomial in $\mathsf {VNP}$ that requires depth-4 circuits of size $2^{\Omega (\sqrt{n}\log n)}$ and Fournier et al. (Proceedings of symposium on theory of computing, ACM, 2014) show that the iterated matrix multiplication polynomial (which is in $\mathsf {VP}$) also requires $2^{\Omega (\sqrt{n}\log n)}$ size depth-4 circuits.

In this paper, we identify a simple combinatorial property such that any polynomial f that satisfies this property would achieve a similar depth-4 circuit-size lower bound. In particular, it does not matter whether f is in $\mathsf {VP}$ or in $\mathsf {VNP}$. 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 $\Omega (n^2)$ for permanent of a $n\times n$ matrix (which is a $n^2$-variate and degree n polynomial) due to Cai et al. (Proceedings of symposium on theory of computing, ACM, 2008). We prove that the determinantal complexity of the iterated matrix multiplication polynomial is $\Omega (dn)$ where d is the number of matrices and n is the dimension of the matrices. In particular, our result settles the determinantal complexity of the iterated matrix multiplication polynomial to $\Theta (dn)$. To the best of our knowledge, a $\Theta (n)$ bound for the determinantal complexity for the iterated matrix multiplication polynomial was known only for any constant $d>1$, due to Jansen (Theory Comput Syst 49(2):343–354, 2011).