Polynomial-Time Approximation Schemes for k-Center and Bounded-Capacity Vehicle Routing in Graphs with Bounded Highway Dimension

The concept of bounded highway dimension was developed to capture observed properties of the metrics of road networks. We show that a graph with bounded highway dimension, for any vertex, can be embedded into a a graph of bounded treewidth in such a way that the distance between $u$ and $v$ is preserved up to an additive error of $\epsilon$ times the distance from $u$ or $v$ to the selected vertex. We show that this theorem yields a PTAS for Bounded-Capacity Vehicle Routing in graphs of bounded highway dimension. In this problem, the input specifies a depot and a set of clients, each with a location and demand; the output is a set of depot-to-depot tours, where each client is visited by some tour and each tour covers at most $Q$ units of client demand. Our PTAS can be extended to handle penalties for unvisited clients. We extend this embedding result to handle a set $S$ of distinguished vertices. The treewidth depends on $|S|$, and the distance between $u$ and $v$ is preserved up to an additive error of $\epsilon$ times the distance from $u$ and $v$ to $S$. This embedding result implies a PTAS for Multiple Depot Bounded-Capacity Vehicle Routing: the tours can go from one depot to another. The embedding result also implies that, for fixed $k$, there is a PTAS for $k$-Center in graphs of bounded highway dimension. In this problem, the goal is to minimize $d$ such that there exist $k$ vertices (the centers) such that every vertex is within distance $d$ of some center. Similarly, for fixed $k$, there is a PTAS for $k$-Median in graphs of bounded highway dimension. In this problem, the goal is to minimize the sum of distances to the $k$ centers.