Metric Approximations of Consistent Path Systems

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.