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).