A (1+ε)-Embedding of Low Highway Dimension Graphs into Bounded Treewidth Graphs

Graphs with bounded highway dimension were introduced by Abraham et al. [Proceedings of SODA 2010, pp. 782--793] as a model of transportation networks. We show that any such graph can be embedded into a distribution over bounded treewidth graphs with arbitrarily small distortion. More concretely, given a weighted graph $G=(V,E)$ of constant highway dimension, we show how to randomly compute a weighted graph $H=(V,E')$ that distorts shortest path distances of $G$ by at most a $1+\varepsilon$ factor in expectation, and whose treewidth is polylogarithmic in the aspect ratio of $G$. Our probabilistic embedding implies quasi-polynomial time approximation schemes for a number of optimization problems that naturally arise in transportation networks, including Travelling Salesman, Steiner Tree, and Facility Location. To construct our embedding for low highway dimension graphs we extend Talwar's [Proceedings of STOC 2004, pp. 281--290] embedding of low doubling dimension metrics into bounded treewidth graphs, whic...

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