Clifford Algebras And Approximating The Permanent

We study approximation algorithms for the permanent of an n x n (0, 1) matrix A based on the following simple idea: obtain a random matrix B by replacing each 1-entry of A independently by +/-e, where e is a random basis element of a suitable algebra; then output \det(B)\(2). This estimator is always unbiased, but it may have exponentially large variance. In our first main result we show that, if we take the algebra to be a Clifford algebra of dimension polynomial in n, then we get an estimator with small variance. Hence, only a constant number of trials suffices to estimate the permanent to good accuracy. The idea of using Clifford algebras is a natural extension of earlier work by Godsil and Gutman, Karmarkar et al., and Barvinok, who used the real numbers, complex numbers and quaternions, respectively. The above result implies that, in principle, this approach gives a fully-polynomial randomized approximation scheme for the permanent, provided \det(B)\(2) can be efficiently computed in the Clifford algebras. Since these algebras are non-commutative it is not clear how to do this. However, our second main result shows how to compute in polynomial time an estimator with the same mean and variance over the 4-dimensional algebra (which is the quaternions, and is non-commutative); in addition to providing some hope that the computations can be performed in higher dimensions, this quaternion algorithm provides an exponential improvement in the variance over that of the 2-dimensional complex version studied by Karmarkar et al. (C) 2003 Elsevier Inc. All rights reserved.