Time-dependent shortest paths in bounded treewidth graphs

TL;DR

This paper proves a bound on the complexity of shortest path arrival functions in graphs with bounded treewidth and introduces an algorithm using star-mesh transformations to compute these functions efficiently.

Contribution

It establishes a tighter bound on the number of breakpoints in arrival functions for bounded treewidth graphs and presents a novel algorithm for their computation.

Findings

Bound on breakpoints: $n^{O(\log^2 w)}$ for graphs with treewidth $w$.

Improved complexity bound over previous work for graphs with unbounded treewidth.

Algorithm for calculating arrival functions using star-mesh transformations.

Abstract

We present a proof that the number of breakpoints in the arrival function between two terminals in graphs of treewidth $w$ is $n^{O(\log^2 w)}$ when the edge arrival functions are piecewise linear. This is an improvement on the bound of $n^{\Theta(\log n)}$ by Foschini, Hershberger, and Suri for graphs without any bound on treewidth. We provide an algorithm for calculating this arrival function using star-mesh transformations, a generalization of the wye-delta-wye transformations.