The Graph of the Pedigree Polytope is Asymptotically Almost Complete (Extended Abstract)

TL;DR

This paper proves that the graph of the pedigree polytope becomes almost complete as the number of cities increases, revealing its asymptotic structural properties related to TSP polytopes.

Contribution

It establishes the asymptotic almost completeness of the pedigree polytope graph, a significant extension of classical TSP polytope graphs, highlighting its unique non-symmetric structure.

Findings

Graph quotient minimum degree tends to 1 as cities increase

Pedigree polytope graph becomes asymptotically almost complete

Differences from symmetric TSP polytope graphs are highlighted

Abstract

Graphs (1-skeletons) of Traveling-Salesman-related polytopes have attracted a lot of attention. Pedigree polytopes are extensions of the classical Symmetric Traveling Salesman Problem polytopes (Arthanari 2000) whose graphs contain the TSP polytope graphs as spanning subgraphs (Arthanari 2013). Unlike TSP polytopes, Pedigree polytopes are not "symmetric", e.g., their graphs are not vertex transitive, not even regular. We show that in the graph of the pedigree polytope, the quotient minimum degree over number of vertices tends to 1 as the number of cities tends to infinity.