An Almost Linear Time Approximation Algorithm for the Permanen of a Random (0-1) Matrix

We present a simple randomized algorithm for approximating permanents. The algorithm with inputs A, ǫ > 0 produces an output XA with (1−ǫ)per(A) ≤ XA ≤ (1+ǫ)per(A) for almost all (0-1) matrices A. For any positive constant ǫ > 0, and almost all (0-1) matrices the algorithm runs in time O(nω), 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(nω) for such matrices. The estimator can also be used to estimate the size of a backtrack tree.