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.

💡 Analysis

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.

📄 Content

Enumeration problems [GJ79] deal with counting the number of solutions in a given instance of a search problem, for example, counting the total number of perfect matchings in a bipartite graph. Their complexity poses unique challenges and surprises. Most of them are NP-hard, and therefore, even if NP = P, it does not imply a polynomial time solution of an NP-hard enumeration problem.

NP-hard enumeration problems fall into a distinct class of polynomial time equivalent problems called the #P-complete problems [Val79b]. As noted by Jerrum [Jer94], problems in #P are ubiquitous-those in FP are more of an exception. What has been found quite surprising is that the enumeration problem for perfect matching in a bipartite graph is #P-complete [Val79a] even though the associated search problem has long been known to be in P [Kuh55,Ege31,Edm65].

Enumeration of a permutation group has long been known to be in FP ( [But91,Hof82]). The basic technique for enumerating a permutation group G (any subgroup of the symmetric group S n ) is based on creating a hierarchy of the Coset Decompositions over a sequence of the subgroups of G, where the smallest subgroup is the trivial group I.

A Coset Decomposition of G is essentially a set of equivalence classes defining a Partition of G for a subgroup H of G, induced by a set of group elements called Coset Representatives(CR). Here each element ψ in CR represents a unique subset of G, called Coset of H in G, obtained by multiplying each element in H by ψ, in certain (right or left) order. For the symmetric group, S n , the partition hierarchy for a fixed subgroup sequence is shown as an n-partite directed acyclic graph in Figure 1, where the nodes in each partition are the elements in CR representing the subsets of a group G (i) in the subgroup sequence G (0) > G (1) • • • > G (n) . The edges represent a disjoint subset relationship. The enumeration technique for perfect matchings extends the above coset decomposition scheme by further partitioning each coset into a family of polynomially many equivalence classes. This extended partition hierarchy (Figure 2) then captures the perfect matchings as an equivalence class in this partition, where each such class allows the P-time enumeration uniformly for all n ≥ 3.

The associated equivalence relation over a coset is induced by a graph theoretic attribute called edge requirements which confirms a potential perfect matching subset in each equivalence class.

The hierarchy of the various classes for a bipartite graph holds the following containment relationship:

The extended partition hierarchy contains “other equivalence classes” CVMPSets and MinSet Sequences, described below.

We map a specific generating set of the symmetric group S n to a graph theoretic “generating set”, such that each coset representative of a (group, subgroup) pair is mapped to a set of graph theoretic coset representatives. This mapping is then used to construct a generating graph for generating all the perfect matchings as directed paths in the generating graph which is a directed acyclic n-partite graph of size O(n 3 ).

Each perfect matching in a bipartite graph with 2n nodes is expressed as a unique directed path of length n-1, called Complete Valid Multiplication Path (CVMP) in the generating graph. The condition for a CVMP of length n-1 to represent a unique perfect matching in the given bipartite graph is captured by an attribute of the CVMP, called Edge Requirement (ER).

The graph theoretic coset representatives induce disjoint subsets of the Cosets, called CVMPSets, an equivalence class containing the CVMPs. Each CVMPSet is further partitioned into polynomially bounded classes called MinSet Sequences induced by the ER of each CVMP, where a MinSet is the set of all Valid Multiplication Paths (VMPs) of common ER.

A judicious choice of the common ER of these VMPs allows a MinSet and any sequence of the MinSets to be P-time enumerable, and which makes the perfect matchings also P-time enumerable as follows.

These MinSet sequences can be viewed as an instance of a perfect matching subproblem, where a sequence containing only one MinSet would represent a set of perfect matchings when each CVMP is of length n-1 with common ER = ∅.

There are exponentially many MinSet sequences in a CVMPSet, and all of them can be covered by only O(n 6 ) unique prefix MinSets. And thus these unique prefixes induce polynomially many equivalence classes each containing exponentially many MinSet sequences (Figure 3). Further, each class size decreases exponentially with n, and thus this hierarchy enables enumeration of all the equivalence classes in polynomial time.

. . . . . . 1. The Mapping : Section 3 develops the concepts leading to the graph theoretic counterparts of the permutation group generating set. It defines the mapping (Lemma 3.5) of a specific generating set of S n to a set of graph theoretic structures.

2. The Extended Partition Hierarchy: Section 4 describ

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