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Revision #1 to TR17-028 | 18th February 2017 20:07

A quadratic lower bound for homogeneous algebraic branching programs

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Revision #1
Authors: Mrinal Kumar
Accepted on: 18th February 2017 20:07
Downloads: 184
Keywords: 


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 $\F[x_1, x_2, \ldots, x_n]$. 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 which computes the polynomial $x_1^n + x_2^n + \ldots + x_n^n$ has at least $\Omega(n^2)$ vertices (and hence edges).

To the best of our knowledge, this seems to be the first non-trivial super-linear lower bound on the number of vertices for a general \emph{homogeneous} ABP and slightly improves the known lower bound of $\Omega(n\log n)$ on the number of edges of in a general (possibly \emph{non-homogeneous}) ABP, which follows from the classical results of Strassen~\cite{Strassen73} and Baur \& Strassen~\cite{BS83}.

On the way, we also get an alternate and essentially unified proof of an $\Omega(n\log n)$ lower bound on the size of a \emph{homogeneous} arithmetic circuit (follows from~\cite{Strassen73, BS83}), and an $n/2$ lower bound ($n$ over $\mathbb{R}$) on the determinantal complexity of an explicit polynomial~\cite{mr04, CCL10, Yabe15}. 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.



Changes to previous version:

Typos and some corrections in proof outline and prior work.


Paper:

TR17-028 | 17th February 2017 17:40

A quadratic lower bound for homogeneous algebraic branching programs





TR17-028
Authors: Mrinal Kumar
Publication: 17th February 2017 18:44
Downloads: 317
Keywords: 


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 $\F[x_1, x_2, \ldots, x_n]$. 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 which computes the polynomial $x_1^n + x_2^n + \ldots + x_n^n$ has at least $\Omega(n^2)$ vertices (and hence edges).

To the best of our knowledge, this seems to be the first non-trivial super-linear lower bound on the number of vertices for a general \emph{homogeneous} ABP and slightly improves the known lower bound of $\Omega(n\log n)$ on the number of edges of in a general (possibly \emph{non-homogeneous}) ABP, which follows from the classical results of Strassen~\cite{Strassen73} and Baur \& Strassen~\cite{BS83}.

On the way, we also get an alternate and essentially unified proof of an $\Omega(n\log n)$ lower bound on the size of a \emph{homogeneous} arithmetic circuit (follows from~\cite{Strassen73, BS83}), and an $n/2$ lower bound ($n$ over $\mathbb{R}$) on the determinantal complexity of an explicit polynomial~\cite{mr04, CCL10, Yabe15}. 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.



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