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
Alternative Algorithms for Counting All Matchings in Graphs
STACS '03 Proceedings of the 20th Annual Symposium on Theoretical Aspects of Computer Science
A determinant-based algorithm for counting perfect matchings in a general graph
SODA '04 Proceedings of the fifteenth annual ACM-SIAM symposium on Discrete algorithms
A polynomial-time approximation algorithm for the permanent of a matrix with nonnegative entries
Journal of the ACM (JACM)
Data streams: algorithms and applications
Foundations and Trends® in Theoretical Computer Science
Solution counting algorithms for constraint-centered search heuristics
CP'07 Proceedings of the 13th international conference on Principles and practice of constraint programming
Non-commutative circuits and the sum-of-squares problem
Proceedings of the forty-second ACM symposium on Theory of computing
An almost linear time approximation algorithm for the permanent of a random (0-1) matrix
FSTTCS'04 Proceedings of the 24th international conference on Foundations of Software Technology and Theoretical Computer Science
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(MATH) 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.(MATH) 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.