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 @math and @math is preserved up to an additive error of @math times the distance from @math or @math 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 @math units of client demand. Our PTAS can be extended to handle penalties for unvisited clients. We extend this embedding result to handle a set @math of distinguished vertices. The treewidth depends on @math , and the distance between @math and @math is preserved up to an additive error of @math times the distance from @math and @math to @math . 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 @math , there is a PTAS for @math -Center in graphs of bounded highway dimension. In this problem, the goal is to minimize @math such that there exist @math vertices (the centers) such that every vertex is within distance @math of some center. Similarly, for fixed @math , there is a PTAS for @math -Median in graphs of bounded highway dimension. In this problem, the goal is to minimize the sum of distances to the @math centers.
* @math - Median We have already surveyed some of the results on @math - Center . Note that the results of Feldmann @cite_9 are based on the definition of highway dimension of 2011 @cite_35 , but can be adapted to the definition we use here.
@cite_35 @cite_9
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We explore the relationship between VC-dimension and graph algorithm design. In particular, we show that set systems induced by sets of vertices on shortest paths have VC-dimension at most two. This allows us to use a result from learning theory to improve time bounds on query algorithms for the point-to-point shortest path problem in networks of low highway dimension, such as road networks. We also refine the definitions of highway dimension and related concepts, making them more general and potentially more relevant to practice. In particular, we define highway dimension in terms of set systems induced by shortest paths, and give cardinality-based and average case definitions. We consider the @math -Center problem and some generalizations. For @math -Center a set of @math center vertices needs to be found in a graph @math with edge lengths, such that the distance from any vertex of @math to its nearest center is minimized. This problem naturally occurs in transportation networks, and therefore we model the inputs as graphs with bounded highway dimension, as proposed by [SODA 2010]. We show both approximation and fixed-parameter hardness results, and how to overcome them using fixed-parameter approximations, where the two paradigms are combined. In particular, we prove that for any @math computing a @math -approximation is W[2]-hard for parameter @math and NP-hard for graphs with highway dimension @math . The latter does not rule out fixed-parameter @math -approximations for the highway dimension parameter @math , but implies that such an algorithm must have at least doubly exponential running time in @math if it exists, unless the ETH fails. On the positive side, we show how to get below the approximation factor of @math by combining the parameters @math and @math : we develop a fixed-parameter @math -approximation with running time @math . Additionally we prove that, unless P=NP, our techniques cannot be used to compute fixed-parameter @math -approximations for only the parameter @math . We also provide similar fixed-parameter approximations for the weighted @math -Center and @math -Partition problems, which generalize @math -Center.