On Approximating the Dynamic and Discrete Network Flow Problem

TL;DR

This paper studies the complexity of dynamic network flow with discrete units, proving APX-hardness in general and providing a PTAS for path graphs with few nodes, along with a new PTAS for a constrained bin packing problem.

Contribution

It establishes the APX-hardness of discrete dynamic network flow and introduces a PTAS for specific graph structures and a novel bin packing variant.

Findings

Dynamic flow with discrete units is APX-hard.

A PTAS exists for path graphs with a constant number of nodes.

A new PTAS is developed for the ready time constrained bin packing problem.

Abstract

We examine the dynamic network flow problem under the assumption that the flow consists of discrete units. The dynamic network flow problem is commonly addressed in the context of developing evacuation plans, where the flow is typically treated as a continuous quantity. However, real-world scenarios often involve moving groups, such as families, as single units. We demonstrate that solving the dynamic flow problem with this consideration is APX-hard. Conversely, we present a PTAS for instances where the base graph is a path with a constant number of nodes. We introduce a `ready time' constraint to the minsum bin packing problem, meaning certain items cannot be placed in specific bins, develop a PTAS for this modified problem, and apply our algorithms to the discrete and dynamic flow problem.