[Paper Review] On the Graph of the Pedigree Polytope

Abdullah Makkeh, Mozhgan Pourmoradnasseri+1 more|arXiv (Cornell University)|Nov 25, 2016

Graph Labeling and Dimension Problems17 references3 citations

TL;DR

This paper investigates the asymptotic structure of the 1-skeleton (graph) of pedigree polytopes, a family of extensions of the symmetric TSP polytope. Using a novel 'adjacency game' model based on Arthanari's polynomial-time adjacency characterization, the authors prove that the minimum vertex degree in the pedigree polytope graph for $n$ cities is asymptotically $(1-o(1)) \cdot (n-1)!/2$, meaning the graph becomes asymptotically almost complete as $n \to \infty$, despite not being regular or symmetric like the TSP polytope.

ABSTRACT

Pedigree polytopes are extensions of the classical Symmetric Traveling Salesman Problem polytopes whose graphs (1-skeletons) contain the TSP polytope graphs as spanning subgraphs. While deciding adjacency of vertices in TSP polytopes is coNP-complete, Arthanari has given a combinatorial (polynomially decidable) characterization of adjacency in Pedigree polytopes. Based on this characterization, we study the graphs of Pedigree polytopes asymptotically, for large numbers of cities. Unlike TSP polytope graphs, which are vertex transitive, Pedigree graphs are not even regular. Using an "adjacency game" to handle Arthanari's intricate inductive characterization of adjacency, we prove that the minimum degree is asymptotically equal to the number of vertices, i.e., the graph is "asymptotically almost complete".

Motivation & Objective

• To understand the asymptotic structure of the 1-skeleton (graph) of pedigree polytopes, which extend the symmetric TSP polytope.
• To analyze the minimum vertex degree in the graph of the pedigree polytope for large $n$, given that adjacency is polynomial-time decidable via Arthanari's characterization.
• To resolve the open question of whether pedigree polytope graphs are asymptotically complete, despite lacking the vertex transitivity of TSP polytopes.
• To develop and apply a novel probabilistic 'adjacency game' model to analyze the evolution of vertex degrees through an inductive construction of the polytope's graph.

Proposed method

• The authors model the graph's evolution using an 'adjacency game' where Alice and Bob select vertices over time, with Bob blindfolded and choosing uniformly at random.
• The game evolves by replacing each vertex with a set of child vertices, governed by rules derived from Arthanari's inductive characterization of adjacency in pedigree polytopes.
• The analysis tracks two key random variables: $T_n$, the number of connected components in the graph at step $n$, and $S_n$, the number of isolated vertices, to bound the probability of low-degree vertices.
• The proof uses concentration bounds and Markov's inequality to show that with high probability, the number of connected components $T_n$ eventually drops to 1 and stays there, implying high minimum degree.
• The authors apply a recursive argument over intervals of increasing size, using conditional probability bounds to control the likelihood of failure in reducing $T_n$, ultimately showing the minimum degree is asymptotically complete.
• The method relies on bounding the probability that $S_n > \ln^2 n$ for any $n$, which would imply low-degree vertices, and shows this probability is negligible for large $n$.

Experimental results

Research questions

• RQ1 Is the graph of the pedigree polytope asymptotically almost complete, meaning its minimum degree approaches the total number of vertices as $n \to \infty$?
• RQ2 Can a probabilistic game model based on Arthanari's adjacency characterization be used to analyze the asymptotic degree distribution in non-symmetric polytopes?
• RQ3 Are there other polytopes, beyond pedigree polytopes, whose graphs are not complete but have minimum degree asymptotically equal to the number of vertices?
• RQ4 Can combinatorial conditions for adjacency in TSP polytopes be derived independently of geometric extensions, potentially yielding graphs closer to the actual TSP polytope?

Key findings

• The minimum degree of the pedigree polytope graph for $n$ cities is asymptotically $(1-o(1)) \cdot (n-1)!/2$ as $n \to \infty$.
• The graph density of the pedigree polytope tends to 1 asymptotically, meaning it is 'asymptotically almost complete'.
• Despite the lack of vertex transitivity, the pedigree polytope graph achieves a minimum degree that is asymptotically equal to the total number of vertices, unlike the TSP polytope graph.
• The probability that the number of connected components $T_n$ exceeds a constant $t$ is bounded by $\varepsilon' / 3$ for large $n$, implying $T_n$ eventually drops to 1 with high probability.
• The probability that the number of isolated vertices $S_n$ exceeds $\ln^2 n$ is at most $\varepsilon$ for sufficiently large $n$, which helps control the likelihood of low-degree vertices.
• The analysis shows that with probability $1 - \varepsilon'$, the number of connected components $T_n$ drops to 1 and remains there for all larger $n$, confirming the asymptotic completeness of the graph.