Abstract
An algebraic branching program (ABP) is a directed acyclic graph, with a start vertex s, and end vertex t and each edge having a weight which is an affine form in variables x_1, x_2, ..., x_n over an underlying field. An ABP computes a polynomial in a natural way, as the sum of weights of all paths from s to t, where the weight of a path is the product of the weights of the edges in the path. An ABP is said to be homogeneous if the polynomial computed at every vertex is homogeneous. In this paper, we show that any homogeneous algebraic branching program which computes the polynomial x_1^n + x_2^n + ... + x_n^n has at least Omega(n^2) vertices (and edges).
To the best of our knowledge, this seems to be the first nontrivial superlinear lower bound on the number of vertices for a general homogeneous ABP and slightly improves the known lower bound of Omega(n log n) on the number of edges in a general (possibly nonhomogeneous) ABP, which follows from the classical results of Strassen (1973) and BaurStrassen (1983).
On the way, we also get an alternate and unified proof of an Omega(n log n) lower bound on the size of a homogeneous arithmetic circuit (follows from [Strassen, 1973] and [BaurStrassen, 1983]), and an n/2 lower bound (n over reals) on the determinantal complexity of an explicit polynomial [MignonRessayre, 2004], [Cai, Chen, Li, 2010], [Yabe, 2015]. These are currently the best lower bounds known for these problems for any explicit polynomial, and were originally proved nearly two decades apart using seemingly different proof techniques.
BibTeX  Entry
@InProceedings{kumar:LIPIcs:2017:7513,
author = {Mrinal Kumar},
title = {{A Quadratic Lower Bound for Homogeneous Algebraic Branching Programs}},
booktitle = {32nd Computational Complexity Conference (CCC 2017)},
pages = {19:119:16},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {9783959770408},
ISSN = {18688969},
year = {2017},
volume = {79},
editor = {Ryan O'Donnell},
publisher = {Schloss DagstuhlLeibnizZentrum fuer Informatik},
address = {Dagstuhl, Germany},
URL = {http://drops.dagstuhl.de/opus/volltexte/2017/7513},
URN = {urn:nbn:de:0030drops75134},
doi = {10.4230/LIPIcs.CCC.2017.19},
annote = {Keywords: algebraic branching programs, arithmetic circuits, determinantal complexity, lower bounds}
}
Keywords: 

algebraic branching programs, arithmetic circuits, determinantal complexity, lower bounds 
Collection: 

32nd Computational Complexity Conference (CCC 2017) 
Issue Date: 

2017 
Date of publication: 

01.08.2017 