On the Importance of Studying the Membership Problem for Pedigree Polytopes

Abstract

Given $n \geq 3$, a combinatorial object called a \textit{ pedigree } is defined using $3$-element subsets from $[n]$ obeying certain conditions. The convex hull of pedigrees is called the pedigree polytope for $n$. Pedigrees are in $1-1$ correspondence with Hamiltonian cycles. Properties of pedigrees, pedigree polytopes, adjacency structure of the graph of the pedigree polytope and their implication on the adjacency structure of the Symmetric Travelling Salesman problem (STSP) polytope have been studied earlier in the literature by the author. The question: Given $X$, does it belong to the pedigree polytope for $n$? is called the membership problem. This article provides proof that the membership problem for pedigree polytopes can be solved efficiently. Due to the pedigree's stem property, we can check the membership problem sequentially for $ k \in [4, n]$. One constructs a layered network, recursively, to check membership in the pedigree polytope. Proof of the proposed framework's validity is given. This article's significant and far-reaching contribution is that the membership problem has a strongly polynomial-time framework. Since the polynomial solvability of the membership problem implies that one can solve efficiently any linear optimisation problem over the pedigree polytope. And a specific linear optimisation over the pedigree polytope (the multistage insertion formulation) solves the STSP. The consequence of this result is that we have proof of $NP = P$. A recent book by the author entitled \textit{Pedigree Polytopes} brings together published results on pedigrees and some new results, mainly in Chapters 5 and 6. The primary purpose of this article is to present the latest results from that book in a self-contained fashion so that experts can vet the same. Some of the proofs and presentation of concepts in this article are new.