The goal of a hub-based distance labeling scheme for a network G = (V, E) is to assign a small subset S(u) @math V to each node u @math V, in such a way that for any pair of nodes u, v, the intersection of hub sets S(u) @math S(v) contains a node on the shortest uv-path. The existence of small hub sets, and consequently efficient shortest path processing algorithms, for road networks is an empirical observation. A theoretical explanation for this phenomenon was proposed by (SODA 2010) through a network parameter they called highway dimension, which captures the size of a hitting set for a collection of shortest paths of length at least r intersecting a given ball of radius 2r. In this work, we revisit this explanation, introducing a more tractable (and directly comparable) parameter based solely on the structure of shortest-path spanning trees, which we call skeleton dimension. We show that skeleton dimension admits an intuitive definition for both directed and undirected graphs, provides a way of computing labels more efficiently than by using highway dimension, and leads to comparable or stronger theoretical bounds on hub set size.
Highway dimension @math guarantees the existence of distance labels of size @math where @math is the weighted diameter of the graph @cite_14 . However, when restricting to polynomial time algorithms, such labels can only be approximated within a @math factor using shortest path cover algorithms @cite_14 or a @math factor with a more involved procedure based on VC-dimension @cite_12 . In any case, this requires an all-pair shortest path computation. For large networks, labels can be practically computed when classical heuristics such as contraction hierarchies (CH) can be performed @cite_14 @cite_19 @cite_22 . Low highway dimension guarantees that there exists an elimination ordering for CH such that the graph produced has bounded size @cite_14 . However, it does not ensure running time faster than all pair shortest path computation.
@cite_19 @cite_14 @cite_22 @cite_12
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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 k-Center problem and some generalizations. For k-Center a set of kcenter vertices needs to be found in a graph G with edge lengths, such that the distance from any vertex ofi¾?G 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 [ICALP 2011]. We show both approximation and fixed-parameter hardness results, and how to overcome them using fixed-parameter approximations. In particular, we prove that for any @math computing a @math -approximation is W[2]-hard for parameter k, and NP-hard for graphs with highway dimension @math . The latter does not rule out fixed-parameter @math -approximations for the highway dimension parameteri¾?h, but implies that such an algorithm must have at least doubly exponential running time in h if it exists, unless the ETH fails. On the positive side, we show how to get below the approximation factor ofi¾?2 by combining the parameters k andi¾?h: we develop a fixed-parameter 3 2-approximation with running time @math . We also provide similar fixed-parameter approximations for the weightedk-Center and @math -Partition problems, which generalize k-Center. The goal of a hub-based distance labeling scheme for a network G = (V, E) is to assign a small subset S(u) ⊆ V to each node u ∈ V, in such a way that for any pair of nodes u, v, the intersection of hub sets S(u) ∩ S(v) contains a node on the shortest uv-path. The existence of small hub sets, and consequently efficient shortest path processing algorithms, for road networks is an empirical observation. A theoretical explanation for this phenomenon was proposed by (SODA 2010) through a network parameter they called highway dimension, which captures the size of a hitting set for a collection of shortest paths of length at least r intersecting a given ball of radius 2r. In this work, we revisit this explanation, introducing a more tractable (and directly comparable) parameter based solely on the structure of shortest-path spanning trees, which we call skeleton dimension. We show that skeleton dimension admits an intuitive definition for both directed and undirected graphs, provides a way of computing labels more efficiently than by using highway dimension, and leads to comparable or stronger theoretical bounds on hub set size. 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.