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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).

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Abstract. Given a complete undirected graph, a cost function on edges, and a degree bound B, the degree bounded network design problem is to find a minimum cost simple subgraph with maximum degree B satisfying given connectivity requirements. Even for a simple connectivity requirement such as finding a spanning tree, computing a feasible solution for the degree bounded network design problem is already NP-hard, and thus there is no polynomial factor approximation algorithm for this problem. In this paper, we show that when the cost function satisfies the triangle inequality, there are constant factor approximation algorithms for various degree bounded network design problems. In global edge-connectivity, there is a (2 + 1 k )-approximation algorithm for the minimum bounded degree k-edge-connected subgraph problem. In local edge-connectivity, there is a 4-approximation algorithm for the minimum bounded degree Steiner network problem when rmax is even, and a 5.5-approximation algorithm when rmax is odd, where rmax is the maximum connectivity requirement. In global vertex-connectivity, there is a (2 + k−1 n + 1 k )-approximation algorithm for the minimum bounded degree k-vertex-connected subgraph problem when n ≥ 2k, where n is the number of vertices. For spanning tree, there is a (1 + 1 B−1 )-approximation algorithm for the minimum bounded degree spanning tree problem. These approximation algorithms return solutions with the smallest possible maximum degree, and in most cases the cost guarantee is obtained by comparing to the optimal cost when there are no degree constraints. This demonstrates that degree constraints can be incorporated into network design problems with metric costs. Our algorithms can be seen as a generalization of Christofides' algorithm for the metric traveling salesman problem. The main technical tool is a simplicity-preserving edge splitting-off operation, which is used to "short-cut" vertices with high degree while maintaining connectivity requirements and preserving simplicity of the solutions. 1. Introduction. Consider finding a minimum cost k-edge-subgraph with maximum degree at most B in a weighted undirected graph. This problem is a generalization of the traveling salesman problem (TSP) when k = B = 2 and the minimum bounded degree spanning tree problem when k = 1. In general this problem does not admit any polynomial time approximation algorithm, since the feasibility problem is already NP-hard. Recent research has thus focused on obtaining bicriteria approximation algorithms for degree bounded network design problems [19,31,39,32].In some network design problems the cost function satisfies the triangle inequality, and stronger algorithmic results are known [27,9,11]. For the TSP, although there is no polynomial factor approximation algorithm in general, it is well known that there is a 1.5-approximation algorithm assuming the triangle inequality [10]. This motivates us to study the more general degree bounded network design problems with metric costs.Formally, we are given a complete ...

Abstract. Given a complete undirected graph, a cost function on edges, and a degree bound B, the degree bounded network design problem is to find a minimum cost simple subgraph with maximum degree B satisfying given connectivity requirements. Even for a simple connectivity requirement such as finding a spanning tree, computing a feasible solution for the degree bounded network design problem is already NP-hard, and thus there is no polynomial factor approximation algorithm for this problem. In this paper, we show that when the cost function satisfies the triangle inequality, there are constant factor approximation algorithms for various degree bounded network design problems. In global edge-connectivity, there is a (2 + 1 k )-approximation algorithm for the minimum bounded degree k-edge-connected subgraph problem. In local edge-connectivity, there is a 4-approximation algorithm for the minimum bounded degree Steiner network problem when rmax is even, and a 5.5-approximation algorithm when rmax is odd, where rmax is the maximum connectivity requirement. In global vertex-connectivity, there is a (2 + k−1 n + 1 k )-approximation algorithm for the minimum bounded degree k-vertex-connected subgraph problem when n ≥ 2k, where n is the number of vertices. For spanning tree, there is a (1 + 1 B−1 )-approximation algorithm for the minimum bounded degree spanning tree problem. These approximation algorithms return solutions with the smallest possible maximum degree, and in most cases the cost guarantee is obtained by comparing to the optimal cost when there are no degree constraints. This demonstrates that degree constraints can be incorporated into network design problems with metric costs. Our algorithms can be seen as a generalization of Christofides' algorithm for the metric traveling salesman problem. The main technical tool is a simplicity-preserving edge splitting-off operation, which is used to "short-cut" vertices with high degree while maintaining connectivity requirements and preserving simplicity of the solutions. 1. Introduction. Consider finding a minimum cost k-edge-subgraph with maximum degree at most B in a weighted undirected graph. This problem is a generalization of the traveling salesman problem (TSP) when k = B = 2 and the minimum bounded degree spanning tree problem when k = 1. In general this problem does not admit any polynomial time approximation algorithm, since the feasibility problem is already NP-hard. Recent research has thus focused on obtaining bicriteria approximation algorithms for degree bounded network design problems [19,31,39,32].In some network design problems the cost function satisfies the triangle inequality, and stronger algorithmic results are known [27,9,11]. For the TSP, although there is no polynomial factor approximation algorithm in general, it is well known that there is a 1.5-approximation algorithm assuming the triangle inequality [10]. This motivates us to study the more general degree bounded network design problems with metric costs.Formally, we are given a complete ...