Metric Approximations of Consistent Path Systems

TL;DR

The paper studies $\alpha$-metric approximations of consistent path systems on graphs, introducing $\Delta(\mathcal{P})$ as the least $\alpha$ for which a path system is $\alpha$-metric with respect to some metric. It constructs, for infinitely many $n$, $n$-point consistent path systems $\mathcal{P}_n$ with $\Delta(\mathcal{P}_n) \ge n^{\frac{1}{2}-o_n(1)}$ using $G$-invariant path systems on Cayley graphs and a Cayley-metric framework. It proves that $\Delta(\mathcal{P})$ is algebraic and computable in polynomial time via LP-based reasoning, and provides a Paley-graph-based construction showing irrational $\Delta$, e.g., $\Delta(\mathcal{P}_{29})$ equals the middle root of $2x^3-3x^2-10x+12$. It concludes with open questions about upper bounds, density of $\Delta$-values, and the computational complexity of the graph parameter $\delta(G)$.

Abstract

A path system $\mathscr{P}$ in a graph $G=(V,E)$ is a collection of paths, with exactly one path between any two vertices in $V$. A path system is said to be consistent if it is closed under subpaths. We say that a path system $\mathscr{P}$ is $α$-metric if there exists a metric $ρ$ on $V$ such that $\sum_{i=1}^{k}ρ(x_{i-1},x_{i}) \le αρ(x_0,x_k)$ for every path $(x_0,x_1,\dots,x_k)\in \mathscr{P}$. Also, we denote by $Δ(\mathscr{P})$ the infimum of $α$ for which $\mathscr{P}$ is $α$-metric. We construct here infinitely many $n$-point consistent path systems $\mathscr{P}_n$ with $Δ(\mathscr{P}_n) \ge n^{\frac{1}{2}-o(1)}$. We also show how to efficiently compute $Δ(\mathscr{P})$ for a given path system.

Figures (1)

• Figure 1: This consistent path system $\mathcal{P}$ in the Petersen graph is non-metric, and yet $\Delta(\mathcal{P})=1$.

Theorems & Definitions (19)

• Definition 2.1
• Definition 2.2
• Definition 2.3
• Proposition 2.4
• proof
• Proposition 2.5
• proof
• Theorem 3.1
• Lemma 3.2
• proof