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

Abstract

AbstractTavenas (Proceedings of mathematical foundations of computer science (MFCS), 2013) has recently proved that any $$n^{O(1)}$$nO(1)-variate and degree n polynomial in $$\\mathsf {VP}$$VP can be computed by a depth-4 \n$$\\Sigma \\Pi \\Sigma \\Pi $$ΣΠΣΠ \ncircuit of size $$2^{O(\\sqrt{n}\\log n)}$$2O(nlogn). So, to prove $$\\mathsf {VP}\\ne \\mathsf {VNP}$$VP≠VNP it is sufficient to show that an explicit polynomial in $$\\mathsf {VNP}$$VNP of degree n requires $$2^{\\omega (\\sqrt{n}\\log n)}$$2ω(nlogn) size depth-4 circuits. Soon after Tavenas’ result, for two different explicit polynomials, depth-4 circuit-size lower bounds of $$2^{\\Omega (\\sqrt{n}\\log n)}$$2Ω(nlogn) 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 $$\\mathsf {VNP}$$VNP that requires depth-4 circuits of size $$2^{\\Omega (\\sqrt{n}\\log n)}$$2Ω(nlogn) and Fournier et al. (Proceedings of symposium on theory of computing, ACM, 2014) show that the iterated matrix multiplication polynomial (which is in $$\\mathsf {VP}$$VP) also requires $$2^{\\Omega (\\sqrt{n}\\log n)}$$2Ω(nlogn) 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 $$\\mathsf {VP}$$VP or in $$\\mathsf {VNP}$$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 $$\\Omega (n^2)$$Ω(n2) for permanent of a $$n\\times n$$n×n matrix (which is a $$n^2$$n2-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 $$\\Omega (dn)$$Ω(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 $$\\Theta (dn)$$Θ(dn). To the best of our knowledge, a $$\\Theta (n)$$Θ(n) bound for the determinantal complexity for the iterated matrix multiplication polynomial was known only for any constant $$d>1$$d>1, due to Jansen (Theory Comput Syst 49(2):343–354, 2011).