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.CCC.2017.20
URN: urn:nbn:de:0030-drops-75217
URL: http://dagstuhl.sunsite.rwth-aachen.de/volltexte/2017/7521/
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Bringmann, Karl ; Ikenmeyer, Christian ; Zuiddam, Jeroen

On Algebraic Branching Programs of Small Width

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LIPIcs-CCC-2017-20.pdf (0.7 MB)

Abstract

In 1979 Valiant showed that the complexity class VP_e of families with polynomially bounded formula size is contained in the class VP_s of families that have algebraic branching programs (ABPs) of polynomially bounded size. Motivated by the problem of separating these classes we study the topological closure VP_e-bar, i.e. the class of polynomials that can be approximated arbitrarily closely by polynomials in VP_e. We describe VP_e-bar with a strikingly simple complete polynomial (in characteristic different from 2) whose recursive definition is similar to the Fibonacci numbers. Further understanding this polynomial seems to be a promising route to new formula lower bounds.

Our methods are rooted in the study of ABPs of small constant width. In 1992 Ben-Or and Cleve showed that formula size is polynomially equivalent to width-3 ABP size. We extend their result (in characteristic different from 2) by showing that approximate formula size is polynomially equivalent to approximate width-2 ABP size. This is surprising because in 2011 Allender and Wang gave explicit polynomials that cannot be computed by width-2 ABPs at all! The details of our construction lead to the aforementioned characterization of VP_e-bar.

As a natural continuation of this work we prove that the class VNP can be described as the class of families that admit a hypercube summation of polynomially bounded dimension over a product of polynomially many affine linear forms. This gives the first separations of algebraic complexity classes from their nondeterministic analogs.

BibTeX - Entry

@InProceedings{bringmann_et_al:LIPIcs:2017:7521,
  author =	{Karl Bringmann and Christian Ikenmeyer and Jeroen Zuiddam},
  title =	{{On Algebraic Branching Programs of Small Width}},
  booktitle =	{32nd Computational Complexity Conference (CCC 2017)},
  pages =	{20:1--20:31},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-040-8},
  ISSN =	{1868-8969},
  year =	{2017},
  volume =	{79},
  editor =	{Ryan O'Donnell},
  publisher =	{Schloss Dagstuhl--Leibniz-Zentrum fuer Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{http://drops.dagstuhl.de/opus/volltexte/2017/7521},
  URN =		{urn:nbn:de:0030-drops-75217},
  doi =		{10.4230/LIPIcs.CCC.2017.20},
  annote =	{Keywords: algebraic branching programs, algebraic complexity theory, border complexity, formula size, iterated matrix multiplication}
}

Keywords: algebraic branching programs, algebraic complexity theory, border complexity, formula size, iterated matrix multiplication
Collection: 32nd Computational Complexity Conference (CCC 2017)
Issue Date: 2017
Date of publication: 01.08.2017

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