License: Creative Commons Attribution 3.0 Unported license (CC BY 3.0)
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DOI: 10.4230/LIPIcs.STACS.2016.46
URN: urn:nbn:de:0030-drops-57475
URL: http://dagstuhl.sunsite.rwth-aachen.de/volltexte/2016/5747/
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Kayal, Neeraj ; Nair, Vineet ; Saha, Chandan

Separation Between Read-once Oblivious Algebraic Branching Programs (ROABPs) and Multilinear Depth Three Circuits

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Abstract

We show an exponential separation between two well-studied models of algebraic computation, namely read-once oblivious algebraic branching programs (ROABPs) and multilinear depth three circuits. In particular we show the following:

1. There exists an explicit n-variate polynomial computable by linear sized multilinear depth three circuits (with only two product gates) such that every ROABP computing it requires 2^{Omega(n)} size.

2. Any multilinear depth three circuit computing IMM_{n,d} (the iterated matrix multiplication polynomial formed by multiplying d, n * n symbolic matrices) has n^{Omega(d)} size. IMM_{n,d} can be easily computed by a poly(n,d) sized ROABP.

3. Further, the proof of 2 yields an exponential separation between multilinear depth four and multilinear depth three circuits: There is an explicit n-variate, degree d polynomial computable by a poly(n,d) sized multilinear depth four circuit such that any multilinear depth three circuit computing it has size n^{Omega(d)}. This improves upon the quasi-polynomial separation result by Raz and Yehudayoff [2009] between these two models.

The hard polynomial in 1 is constructed using a novel application of expander graphs in conjunction with the evaluation dimension measure used previously in Nisan [1991], Raz [2006,2009], Raz and Yehudayoff [2009], and Forbes and Shpilka [2013], while 2 is proved via a new adaptation of the dimension of the partial derivatives measure used by Nisan and Wigderson [1997]. Our lower bounds hold over any field.

BibTeX - Entry

@InProceedings{kayal_et_al:LIPIcs:2016:5747,
  author =	{Neeraj Kayal and Vineet Nair and Chandan Saha},
  title =	{{Separation Between Read-once Oblivious Algebraic Branching Programs (ROABPs) and Multilinear Depth Three Circuits}},
  booktitle =	{33rd Symposium on Theoretical Aspects of Computer Science (STACS 2016)},
  pages =	{46:1--46:15},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-001-9},
  ISSN =	{1868-8969},
  year =	{2016},
  volume =	{47},
  editor =	{Nicolas Ollinger and Heribert Vollmer},
  publisher =	{Schloss Dagstuhl--Leibniz-Zentrum fuer Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{http://drops.dagstuhl.de/opus/volltexte/2016/5747},
  URN =		{urn:nbn:de:0030-drops-57475},
  doi =		{10.4230/LIPIcs.STACS.2016.46},
  annote =	{Keywords: multilinear depth three circuits, read-once oblivious algebraic branching programs, evaluation dimension, skewed partial derivatives, expander graphs,}
}

Keywords: multilinear depth three circuits, read-once oblivious algebraic branching programs, evaluation dimension, skewed partial derivatives, expander graphs,
Collection: 33rd Symposium on Theoretical Aspects of Computer Science (STACS 2016)
Issue Date: 2016
Date of publication: 16.02.2016

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