An almost linear time approximation algorithm for the permanent of a random (0-1) matrix

We present a simple randomized algorithm for approximating permanents. The algorithm with inputs A, ∈ > 0 produces an output XA with (1 - ∈)per(A) ≤ X_A ≤ (1 + ∈)per(A) for almost all (0-1) matrices A. For any positive constant ∈ > 0, and almost all (0-1) matrices the algorithm runs in time O(n²ω), i.e., almost linear in the size of the matrix, where ω = ω(n) is any function satisfying ω(n) → ∞ as n → ∞. This improves the previous bound of O(n³ω) for such matrices. The estimator can also be used to estimate the size of a backtrack tree.