Conductance and the rapid mixing property for Markov chains: the approximation of permanent resolved

The permanent of an n x n matrix A with 0-1 entries aij is defined by per (A) = Σ/σ Π/n-1/i=ο aiσ(i), where the sum is over all permutations σ of [n] = {0, …, n - 1}. Evaluating per (A) is equivalent to counting perfect matchings (1-factors) in the bipartite graph G = (V1, V2, E), where V1 = V2 = [n] and (i,j) ∈ E iff aij = 1. The permanent function arises naturally in a number of fields, including algebra, combinatorial enumeration and the physical sciences, and has been an object of study by mathematicians for many years (see [14] for background). Despite considerable effort, and in contrast with the syntactically very similar determinant, no efficient procedure for computing this function is known.