How hard is it to marry at random? (On the approximation of the permanent)
STOC '86 Proceedings of the eighteenth annual ACM symposium on Theory of computing
SIAM Journal on Computing
Lower bounds for non-commutative computation
STOC '91 Proceedings of the twenty-third annual ACM symposium on Theory of computing
A Monte-Carlo algorithm for estimating the permanent
SIAM Journal on Computing
Random Structures & Algorithms
A polynomial-time approximation algorithm for the permanent of a matrix with non-negative entries
STOC '01 Proceedings of the thirty-third annual ACM symposium on Theory of computing
On approximating the permanent and other #p-complete problems
On approximating the permanent and other #p-complete problems
On determinant-based algorithms for counting perfect matchings in graphs
On determinant-based algorithms for counting perfect matchings in graphs
On the hardness of the noncommutative determinant
Proceedings of the forty-second ACM symposium on Theory of computing
Almost settling the hardness of noncommutative determinant
Proceedings of the forty-third annual ACM symposium on Theory of computing
Approximate counting of cycles in streams
ESA'11 Proceedings of the 19th European conference on Algorithms
Approximating the Permanent via Nonabelian Determinants
SIAM Journal on Computing
Noncommutativity makes determinants hard
ICALP'13 Proceedings of the 40th international conference on Automata, Languages, and Programming - Volume Part I
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We study approximation algorithms for the permanent of an n × 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 noncommutative 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.