Graph Orientation and Flows Over Time

TL;DR

This paper investigates how the orientation of undirected networks affects flow over time, establishing bounds on flow loss due to orientation and analyzing the complexity of optimizing orientations.

Contribution

It introduces the concept of the price of orientation, proves tight bounds on flow retention, and explores the computational hardness of finding optimal orientations.

Findings

Always an orientation exists with at least 1/3 of the flow retained.

For single source or sink networks, at least 1/2 of the flow can be preserved.

Finding the optimal orientation is computationally hard (non-approximable).

Abstract

Flows over time are used to model many real-world logistic and routing problems. The networks underlying such problems -- streets, tracks, etc. -- are inherently undirected and directions are only imposed on them to reduce the danger of colliding vehicles and similar problems. Thus the question arises, what influence the orientation of the network has on the network flow over time problem that is being solved on the oriented network. In the literature, this is also referred to as the contraflow or lane reversal problem. We introduce and analyze the price of orientation: How much flow is lost in any orientation of the network if the time horizon remains fixed? We prove that there is always an orientation where we can still send $\frac{1}{3}$ of the flow and this bound is tight. For the special case of networks with a single source or sink, this fraction is $\frac12$ which is again tight. We present more results of similar flavor and also show non-approximability results for finding the best orientation for single and multicommodity maximum flows over time.