The Collapse of the Polynomial Hierarchy: #P = FP

We present a novel extension to the permutation group enumeration technique which is well known to have polynomial time algorithms. We use this extended technique to count all the perfect matchings in a bipartite graph. We further show how this technique can be used for solving a class of #P-complete counting problems by NC-algorithms, where the solution space of the associated search problems spans a symmetric group. Two examples of the natural candidates in this class are Perfect Matching and Hamiltonian Circuit problems. The sequential time complexity and the parallel (NC) processor complexity of counting all the solutions to these two problems are shown to be O(n 19 log(n)) and O(n 19) respectively. Hence we prove a result even more surprising than NP = P, that is, #P = FP. By the main Theorem of Toda (IEEE FOCS, 1989) this result concludes the collapse of the Polynomial Hierarchy.