Flows over time in time-varying networks

TL;DR

This paper advances the theory of network flows over time by developing continuous time analogues of key static flow concepts, including optimality conditions and duality, to improve understanding of dynamic network flow models.

Contribution

It introduces continuous time versions of fundamental static network flow concepts such as optimality conditions and strong duality, filling gaps in the theoretical framework.

Findings

Established a reduced cost optimality condition for continuous time flows

Developed a negative cycle optimality condition in the continuous setting

Proved a strong duality theorem for a broad class of time-varying network flows

Abstract

There has been much research on network flows over time due to their important role in real world applications. This has led to many results, but the more challenging continuous time model still lacks some of the key concepts and techniques that are the cornerstones of static network flows. The aim of this paper is to advance the state of the art for dynamic network flows by developing the continuous time analogues of the theory for static network flows. Specifically, we make use of ideas from the static case to establish a reduced cost optimality condition, a negative cycle optimality condition, and a strong duality result for a very general class of network flows over time.