The k-interchange-constrained diameter of a transit network: a connectedness indicator that accounts for travel convenience

ABSTRACT

We study two variants of the shortest path problem. Given an integer $k$, the $k$-color-constrained and the $k$-interchange-constrained shortest path problems, respectively, seek a shortest path that uses no more than $k$ colors and one that makes no more than $k-1$ alternations of colors. We show that the former problem is NP-hard, when the latter is tractable. The study of these problems is motivated by some limitations in the use of diameter-based metrics to evaluate the topological structure of transit networks. We notably show that indicators such as the diameter or directness of a transit network fail to adequately account for travel convenience in measuring the connectivity of a network and propose a new network indicator, based on solving the $k$-interchange-constrained shortest path problem, that aims at alleviating these limitations.

KEYWORDS:

• Graph theory
• shortest path problem
• computational complexity
• transit networks
• network indicators

Acknowledgments

The author would like to thank Dr. Yuval Filmus for his helpful advice, as well as two anonymous reviewers for their valuable feedback that greatly improved this paper.

Disclosure statement

No potential conflict of interest was reported by the author.

Notes

1. In transit networks, intermediate nodes $x_i,i=1,\dots ,k-1$ would necessarily be interchange nodes.