Cancellation is exponentially powerful for computing the determinant

Abstract

By defining a notion of cancellation in algebraic circuits, we study the power of negation at a finer granularity than previously considered. Our result is that in the absence of cancellation, computing the determinant requires exponential circuit size. Previous work shows that cancellations are essential for efficiently computing certain monotone (all coefficients positive) polynomials. The work presented in this paper extends that to the computation of non-monotone polynomials and shows that it is the cancellative aspect of negation that allows efficient computation of even certain non-monotone polynomials. We present the first example of a non-monotone polynomial for which cancellations lead to exponential savings in circuit size: determinant.