Derandomizing the Isolation Lemma and Lower Bounds for Circuit Size

  • Authors:

  • V. Arvind;Partha Mukhopadhyay

  • Affiliations:

  • Institute of Mathematical Sciences, , Chennai, India 600 113;Institute of Mathematical Sciences, , Chennai, India 600 113

  • Venue:
  • APPROX '08 / RANDOM '08 Proceedings of the 11th international workshop, APPROX 2008, and 12th international workshop, RANDOM 2008 on Approximation, Randomization and Combinatorial Optimization: Algorithms and Techniques
  • Year:
  • 2008

Quantified Score

Hi-index 0.00

Visualization

Abstract

The isolation lemma of Mulmuley et al [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 deterministicidentity tests has received a lot of attention recently. The goal of this paper is to compare the isolation lemma with polynomial identity testing:1We 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.1From the result of Klivans-Spielman [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 or the Permanent over does not have polynomial-size arithmetic circuits.