Please note that the content of this book primarily consists of articles available from Wikipedia or other free sources online. In a 1979 paper Leslie Valiant prove that the problem of computing the permanent of a matrix is #P-hard, and remains #P-complete even if the matrix is restricted to have entries that are all 0 or 1. This result is sometimes known as Valiant's theorem and is considered a seminal result in computational complexity theory.Specifically, computing the permanent (shown to be difficult by Valiant's results) is closely connected with finding a perfect matchingin a bipartite graph, which is solvable in polynomial time by the Hopcroft-Karp algorithm. For a bipartite graph with 2n vertices partitioned into two parts with n vertices each, the number of perfect matchings equals the permanent of its biadjacency matrix and the square of the number of perfect matchings is equal to the permanent of its adjacency matrix.