Exact and Approximate Hierarchical Hub Labeling

Hub Labeling (HL) is a state-of-the-art technique for accelerating shortest path computation in road networks. By utilizing precomputed node labels, it can answer distance queries in microseconds on continent-sized networks. The optimization goal is to get correct query results with a minimum number of labels. There is an \(\mathcal {O}(\log n)\) approximation algorithm for the size of an HL with a running time of \(\mathcal {O}(n^3\log n)\) . However, existing practical implementations rely mostly on heuristics for a special type of HL, so called Hierarchical HL (HHL). Deciding whether a graph admits a labeling of size at most k is NP-hard for both HL and HHL. For HHL, an \(\mathcal {O}(\sqrt{n} \log n)\) approximation algorithm (called w-HHL) is known, as well as a parametrized upper bound of \(\mathcal {O}(t \log n)\) on the average label size, where t denotes the treewidth of the network. In this paper, we devise an exact HHL algorithm for general graphs. Furthermore, we improve the parametrized approximation factor to \(\mathcal {O}(b)\) , where b denotes the balanced separator number with \(b-1 \le t\) . We also show that w-HHL provides a constant factor approximation on trees, and investigate for the first time the practical performance of existing HHL approximation algorithms. Our theoretical results offer some explanatory power for the good performance of HHL on road networks, while our experimental results show that commonly used methods for HHL are noticeably outperformed by w-HHL on general graphs as well as trees.