Metric Approximations of Consistent Path Systems

Abstract

A path system $\mathcal{P}$ 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 $\mathcal{P}$ is $\hat{I}\pm$-metric if there exists a metric $\ddot{I}$ on $V$ such that ${\sum }_{i=1}^k\ddot{I}(x_{i-1},x_i)\le \hat{I}\pm \ddot{I}(x_0,x_k)$ for every path $(x_0,x_1,\dots ,x_k)\in \mathcal{P}$. Also, we denote by $\hat{I}”(\mathcal{P})$ the infimum of $\hat{I}\pm$ for which $\mathcal{P}$ is $\hat{I}\pm$-metric. We show that $\hat{I}”(\mathcal{P})\le O(\sqrt{n})$ for every $n$-point consistent path system $\mathcal{P}$. On the other hand, we construct infinitely many $n$-point consistent path systems ${\mathcal{P}}_n$ with $\hat{I}”({\mathcal{P}}_n)\ge \tilde{\hat{I}}\text{c◯}(\sqrt{n})$, showing these bounds are tight up to a polylogarithmic factor. We also show how to efficiently compute $\hat{I}”(\mathcal{P})$ for a given path system.

15 pages