An Extension of the Permutation Group Enumeration Technique (Collapse of the Polynomial Hierarchy: $\mathbf{NP = P}$)

Abstract

The distinguishing result of this paper is a $\mathbf{P}$-time enumerable partition of all the potential perfect matchings in a bipartite graph. This partition is a set of equivalence classes induced by the missing edges in the potential perfect matchings. We capture the behavior of these missing edges in a polynomially bounded representation of the exponentially many perfect matchings by a graph theoretic structure, called MinSet Sequence, where MinSet is a P-time enumerable structure derived from a graph theoretic counterpart of a generating set of the symmetric group. This leads to a polynomially bounded generating set of all the classes, enabling the enumeration of perfect matchings in polynomial time. The sequential time complexity of this $\mathbf{\#P}$-complete problem is shown to be $O(n^{45}\log n)$. And thus we prove a result even more surprising than $\mathbf{NP = P}$, that is, $\mathbf{\#P}=\mathbf{FP}$, where $\mathbf{FP}$ is the class of functions, $f: \{0, 1\}^* \rightarrow \mathbb{N} $, computable in polynomial time on a deterministic model of computation.