Revision #3 Authors: Vikraman Arvind, Partha Mukhopadhyay

Accepted on: 16th August 2008 00:00

Downloads: 2771

Keywords:

The isolation lemma of Mulmuley et al cite{MVV87} is an important

tool in the design of randomized algorithms and has played an

important role in several nontrivial complexity upper bounds. On

the other hand, polynomial identity testing is a well-studied

algorithmic problem with efficient randomized algorithms and the

problem of obtaining efficient emph{deterministic} identity tests

has received a lot of attention recently. The goal of this note is

to compare the isolation lemma with polynomial identity testing:

begin{enumerate}

item We show that derandomizing reasonably restricted versions of the

isolation lemma implies circuit size lower bounds. We derive the

circuit lower bounds by examining the connection between the

isolation lemma and polynomial identity testing. We give a

randomized polynomial-time identity test for noncommutative circuits

of polynomial degree based on the isolation lemma. Using this

result, we show that derandomizing the isolation lemma implies

noncommutative circuit size lower bounds. For the commutative case,

a stronger derandomization hypothesis allows us to construct an

explicit multilinear polynomial that does not have subexponential

size commutative circuits. The restricted versions of the isolation

lemma we consider are natural and would suffice for the standard

applications of the isolation lemma.

item From the result of Klivans-Spielman cite{KS01} we observe that

there is a randomized polynomial-time identity test for commutative

circuits of polynomial degree, also based on a more general

isolation lemma for linear forms. Consequently, derandomization of

(a suitable version of) this isolation lemma implies that either

$NEXPnotsubset P/poly$ or the Permanent over $Z$ does not have

polynomial-size arithmetic circuits.

end{enumerate}

Revision #2 Authors: Vikraman Arvind, Partha Mukhopadhyay

Accepted on: 2nd May 2008 00:00

Downloads: 2820

Keywords:

The isolation lemma of Mulmuley et al \cite{MVV87} is an important

tool in the design of randomized algorithms and has played an

important role in several nontrivial complexity upper bounds. On

the other hand, polynomial identity testing is a well-studied

algorithmic problem with efficient randomized algorithms and the

problem of obtaining efficient \emph{deterministic} identity tests

has received a lot of attention recently. The goal of this note is

to compare the isolation lemma with polynomial identity testing:

\begin{enumerate}

\item We show that derandomizing reasonably restricted versions of the

isolation lemma implies circuit size lower bounds. We derive the

circuit lower bounds by examining the connection between the

isolation lemma and polynomial identity testing. We give a

randomized polynomial-time identity test for noncommutative circuits

of polynomial degree based on the isolation lemma. Using this

result, we show that derandomizing the isolation lemma implies

noncommutative circuit size lower bounds. For the commutative case,

a stronger derandomization hypothesis allows us to construct an

explicit multilinear polynomial that does not have subexponential

size commutative circuits. The restricted versions of the isolation

lemma we consider are natural and would suffice for the standard

applications of the isolation lemma.

\item From the result of Klivans-Spielman \cite{KS01} we observe that

there is a randomized polynomial-time identity test for commutative

circuits of polynomial degree, also based on a more general

isolation lemma for linear forms. Consequently, derandomization of

(a suitable version of) this isolation lemma implies that either

$\NEXP\not\subset \P/\poly$ or the Permanent over $\Z$ does not have

polynomial-size arithmetic circuits.

\end{enumerate}

Revision #1 Authors: Vikraman Arvind, Partha Mukhopadhyay

Accepted on: 30th April 2008 00:00

Downloads: 2995

Keywords:

The isolation lemma of Mulmuley et al \cite{MVV87} is an

important tool in the design of randomized algorithms and has played an

important role in several nontrivial complexity upper bounds. On

the other hand, polynomial identity testing is a well-studied

algorithmic problem with efficient randomized algorithms and the

problem of obtaining efficient \emph{deterministic} identity tests

has received a lot of attention recently. The goal of this note is

to compare the isolation lemma with polynomial identity testing:

1. We show that derandomizing reasonably restricted versions of the

isolation lemma implies circuit size lower bounds. We derive the

circuit lower bounds by examining the connection between the

isolation lemma and polynomial identity testing. We give a

randomized polynomial-time identity test for noncommutative circuits

of polynomial degree based on the isolation lemma. Using this

result, we show that derandomizing the isolation lemma implies

noncommutative circuit size lower bounds. The restricted versions of

the isolation lemma we consider are natural and would suffice for

the standard applications of the isolation lemma.

2. From the result of Klivans-Spielman \cite{KS01} we observe that

there is a randomized polynomial-time identity test for commutative

circuits of polynomial degree, also based on a more general

isolation lemma for linear forms. Consequently, derandomization of

(a suitable version of) this isolation lemma implies that either

$\NEXP\not\subset \P/\poly$ or the Permanent over $\Z$ does not have

polynomial-size arithmetic circuits.

TR08-049 Authors: Vikraman Arvind, Partha Mukhopadhyay

Publication: 29th April 2008 20:49

Downloads: 3080

Keywords:

We give a randomized polynomial-time identity test for

noncommutative circuits of polynomial degree based on the isolation

lemma. Using this result, we show that derandomizing the isolation

lemma implies noncommutative circuit size lower bounds. More

precisely, we consider two restricted versions of the isolation

lemma and show that derandomizing each of them implies nontrivial

circuit size lower bounds for noncommutative circuits. These

restricted versions of the isolation lemma are natural and would

suffice for the standard applications of the isolation lemma.