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

摘要

Graphs with bounded highway dimension were introduced in [Abraham et al., SODA 2010] 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, if the highway dimension of G is constant we show how to randomly compute a subgraph of the shortest path metric of the input graph G with the following two properties: it distorts the distances of G by a factor of 1 + ε in expectation and has a treewidth that is poly-logarithmic in the aspect ratio of G. In particular, this result 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 [STOC 2004] embedding of low doubling dimension metrics into bounded treewidth graphs, which generalizes known results for Euclidean metrics. We add several non-trivial ingredients to Talwar's techniques, and in particular thoroughly analyze the structure of low highway dimension graphs. Thus we demonstrate that the geometric toolkit used for Euclidean metrics extends beyond the class of low doubling metrics.