Abstract

Let A be an n \Theta n matrix with 0-1 valued entries, and let per(A) be the permanent of A. We describe a Monte-Carlo algorithm which produces a "good in the relative sense" estimate of per(A) and has running time poly(n)2 n=2 , where poly(n) denotes a function that grows polynomially with n. 1 Introduction Let A be an n \Theta n matrix with 0-1 valued entries, let det(A) denote the determinant of A and let per(A) denote the permanent of A. The marked contrast between the computational complexity of computing det(A) versus that of computing per(A), despite the deceiving similarity between the two tasks, has baffled researchers for years. One of the reasons for interest in computing per(A) is that A can be viewed as the adjacency matrix of a bipartite graph, H = (X; Y; E) where X corresponds 1 to the rows in A, Y to the columns in A, and A ij = 1 if there is and edge between X i and Y j . The quantity per(A) is exactly the number of perfect matchings in H. It is well known tha...

Citations