Clifford algebras and approximating the permanent
Journal of Computer and System Sciences - STOC 2002
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We present a collection of results on approximately counting the number of perfect matchings in a graph by computing the determinants of related random matrices. We first study approximation algorithms for the permanent of an n × n (0,1) matrix A, a problem equivalent to counting perfect matchings in a bipartite graph. Our algorithms are based on the following simple idea, which goes back to Godsil and Gutman: 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 critical ratio. (The critical ratio of an estimator is the ratio of its second moment to the square of its mean, and governs how many samples of the estimator are needed to obtain a good approximation with high probability.) 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 constant critical ratio. 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 unbiased estimator with the same critical ratio 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 critical ratio over that of the 2-dimensional complex version studied by Karmarkar et al. We then investigate a wide variety of approaches to extending this algorithm to higher-dimensional Clifford algebras and evaluate their potential for success. Finally, we extend this approach to the problem of counting perfect matchings in general (non-bipartite) graphs. Here, for a graph G, our algorithm constructs a random matrix B based on the Tutte matrix of G, and then outputs det(B). This gives us an unbiased estimator that is a natural generalization of the 1-dimensional Godsil-Gutman estimator for bipartite graphs, and we show that it shares the same upper bound on its critical ratio.