A Characterization of Polynomial Time Enumeration (Collapse of the Polynomial Hierarchy: $\mathbf{NP = P}$)
نویسنده
چکیده
We resolve the NP = P? question by providing an existential proof to the following conjecture on the characterization of polynomial time enumeration: A sufficient condition for the existence of a P-time algorithm for any enumeration problem is the existence of a partition hierarchy of the exponentially decreasing solution spaces, where each partition is polynomially bounded and the disjoint subsets in each partition are P-time enumerable for each n ≥ 1, n being a problem size parameter. The existential proof is a P-time counting algorithm for the perfect matchings, obtained by extending the basic enumeration technique for permutation groups to the set of perfect matchings in a bipartite graph. The sequential time complexity of this #P-complete problem is shown to be O(n logn). 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 such as a deterministic Turing machine or a RAM.
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