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Graphs with bounded highway dimension were introduced by 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, 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 + ε 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 [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 analyse 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.following formal definition, if dist(u, v) denotes the shortest-path distance between vertices u and v, let B r (v) = {u ∈ V |dist(u, v) ≤ r} be the ball of radius r centred at v. We will also say that a path P lies inside B r (v) if all its vertices lie inside B r (v).Definition 1.1. The highway dimension of a graph G is the smallest integer k such that, for some universal constant c ≥ 4, for every r ∈ R + , and every ball B cr (v) of radius cr, there are at most k vertices in B cr (v) hitting all shortest paths of length more than r that lie in B cr (v).Rather than working with the above definition directly, we often consider the closely related notion of shortest path covers (also introduced in [1]).Definition 1.2. For a graph G and r ∈ R + , a shortest path cover spc(r) ⊆ V is a set of hubs that intersect all shortest paths of length in (r, cr/2] of G. Such a cover is called locally s-sparse for scale r, if no ball of radius cr/2 contains more than s vertices from spc(r).

We prove that no online algorithm (even randomized, against an oblivious adversary) is better than 1/2-competitive for welfare maximization with coverage valuations, unless N P = RP . Since the Greedy algorithm is known to be 1/2-competitive for monotone submodular valuations, of which coverage is a special case, this proves that Greedy provides the optimal competitive ratio. On the other hand, we prove that Greedy in a stochastic setting with i.i.d. items and valuations satisfying diminishing returns is (1 − 1/e)-competitive, which is optimal even for coverage valuations, unless N P = RP . For online budget-additive allocation, we prove that no algorithm can be 0.612-competitive with respect to a natural LP which has been used previously for this problem.