M ar 2 00 9 The Collapse of the Polynomial Hierarchy : NP = P ( A Summary )

We present a novel extension to the permutation group enumeration technique which is well known to have polynomial time algorithms. This extended technique allows each perfect matching in a bipartite graph on 2n nodes to be expressed as a unique directed path in a directed acyclic graph of size O(n). Thus it transforms the perfect matching counting problem into a directed path counting problem for directed acyclic graphs. 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 the above two counting problems are shown to be O(n log n) and O(n) respectively. And thus we prove a result even more surprising than NP = P, that is, #P = FP, where FP is the class of functions, f : {0, 1}∗ → N, computable in polynomial time on a deterministic model of computation. It is well established that NP ⊆ P, and hence the Polynomial Time Hierarchy collapses to P.