Metric Approximations of Consistent Path Systems

TL;DR

This paper introduces a new class of consistent path systems in graphs, explores their metric approximation properties, and provides methods to compute their metric distortion efficiently.

Contribution

It constructs infinitely many consistent path systems with high metric distortion and presents an efficient algorithm for computing the metric approximation parameter.

Findings

Constructed path systems with distortion at least n^{1/2 - o(1)}

Established methods to compute the metric distortion parameter efficiently

Provided theoretical bounds on the metric approximation of consistent path systems

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 $\alpha$-metric if there exists a metric $\rho$ on $V$ such that $\sum_{i=1}^{k}\rho(x_{i-1},x_{i}) \le \alpha \rho(x_0,x_k)$ for every path $(x_0,x_1,\dots,x_k)\in \mathscr{P}$. Also, we denote by $\Delta(\mathscr{P})$ the infimum of $\alpha$ for which $\mathscr{P}$ is $\alpha$-metric. We construct here infinitely many $n$-point consistent path systems $\mathscr{P}_n$ with $\Delta(\mathscr{P}_n) \ge n^{\frac{1}{2}-o(1)}$. We also show how to efficiently compute $\Delta(\mathscr{P})$ for a given path system.