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, ε> 0 produces an output XA with (1 − ε)per(A) ≤ XA ≤ (1 + ε) per (A) for almost all (0-1) matrices A. For any positive constant ε > 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.