Lower Bounds in the Preprocessing and Query Phases of Routing Algorithms

TL;DR

This paper establishes lower bounds for query times and preprocessing steps in popular routing algorithms, providing insights into their theoretical limitations on real-world graphs.

Contribution

It presents the first lower bounds for query times in contraction hierarchies, transit node routing, and hub labeling, and proves NP-hardness of optimal preprocessing for hub labeling.

Findings

Lower bounds for query times in three routing algorithms.

Improved lower bounds on the number of shortcuts in contraction hierarchies.

NP-hardness proof for optimal preprocessing in hub labeling.

Abstract

In the last decade, there has been a substantial amount of research in finding routing algorithms designed specifically to run on real-world graphs. In 2010, Abraham et al. showed upper bounds on the query time in terms of a graph's highway dimension and diameter for the current fastest routing algorithms, including contraction hierarchies, transit node routing, and hub labeling. In this paper, we show corresponding lower bounds for the same three algorithms. We also show how to improve a result by Milosavljevic which lower bounds the number of shortcuts added in the preprocessing stage for contraction hierarchies. We relax the assumption of an optimal contraction order (which is NP-hard to compute), allowing the result to be applicable to real-world instances. Finally, we give a proof that optimal preprocessing for hub labeling is NP-hard. Hardness of optimal preprocessing is known for most routing algorithms, and was suspected to be true for hub labeling.