License: Creative Commons Attribution 3.0 Unported license (CC-BY 3.0)
When quoting this document, please refer to the following
DOI: 10.4230/LIPIcs.STACS.2014.239
URN: urn:nbn:de:0030-drops-44610
URL: https://drops.dagstuhl.de/opus/volltexte/2014/4461/
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Chillara, Suryajith ; Mukhopadhyay, Partha

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

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Abstract

Tavenas has recently proved that any n^{O(1)}-variate and degree n polynomial in VP can be computed by a depth-4 SigmaPi^[O(sqrt{n})]SigmaPi^{[sqrt{n}]} circuit of size 2^{O(n^{1/2}.log(n))} [Tavenas, 2013]. So, to prove that VP is not equal to VNP it is sufficient to show that an explicit polynomial in VNP of degree n requires 2^{omega(n^{1/2}.log(n))} size depth-4 circuits. Soon after Tavenas' result, for two different explicit polynomials, depth-4 circuit size lower bounds of 2^{Omega(n^{1/2}.log(n))} have been proved (see [Kayal, Saha, and Saptharishi, 2013] and [Fournier et al., 2013]). In particular, using combinatorial design [Kayal et al., 2013] construct an explicit polynomial in VNP that requires depth-4 circuits of size 2^{Omega(n^{1/2}.log(n))} and [Fournier et al., 2013] show that the iterated matrix multiplication polynomial (which is in VP) also requires 2^{Omega(n^{1/2}.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 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 Omega(n^2) for Permanent of a nxn matrix (which is a n^2-variate and degree n polynomial) [Cai, Chen, and Li, 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. 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 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 [Jansen, 2011].

BibTeX - Entry

@InProceedings{chillara_et_al:LIPIcs:2014:4461,
  author =	{Suryajith Chillara and Partha Mukhopadhyay},
  title =	{{Depth-4 Lower Bounds, Determinantal Complexity: A Unified Approach}},
  booktitle =	{31st International Symposium on Theoretical Aspects of Computer Science (STACS 2014)},
  pages =	{239--250},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-939897-65-1},
  ISSN =	{1868-8969},
  year =	{2014},
  volume =	{25},
  editor =	{Ernst W. Mayr and Natacha Portier},
  publisher =	{Schloss Dagstuhl--Leibniz-Zentrum fuer Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{http://drops.dagstuhl.de/opus/volltexte/2014/4461},
  URN =		{urn:nbn:de:0030-drops-44610},
  doi =		{10.4230/LIPIcs.STACS.2014.239},
  annote =	{Keywords: Arithmetic Circuits, Determinantal Complexity, Depth-4 Lower Bounds}
}

Keywords: Arithmetic Circuits, Determinantal Complexity, Depth-4 Lower Bounds
Collection: 31st International Symposium on Theoretical Aspects of Computer Science (STACS 2014)
Issue Date: 2014
Date of publication: 05.03.2014


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