de.mpg.escidoc.pubman.appbase.FacesBean
English
 
Help Guide Disclaimer Contact us Login
  Advanced SearchBrowse

Item

ITEM ACTIONSEXPORT

Released

Paper

On Algebraic Branching Programs of Small Width

MPS-Authors
http://pubman.mpdl.mpg.de/cone/persons/resource/persons44182

Bringmann,  Karl
Algorithms and Complexity, MPI for Informatics, Max Planck Society;

http://pubman.mpdl.mpg.de/cone/persons/resource/persons202366

Ikenmeyer,  Christian
Algorithms and Complexity, MPI for Informatics, Max Planck Society;

Locator
There are no locators available
Fulltext (public)

arXiv:1702.05328.pdf
(Preprint), 443KB

Supplementary Material (public)
There is no public supplementary material available
Citation

Bringmann, K., Ikenmeyer, C., & Zuiddam, J. (2017). On Algebraic Branching Programs of Small Width. Retrieved from http://arxiv.org/abs/1702.05328.


Cite as: http://hdl.handle.net/11858/00-001M-0000-002D-89A4-8
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.