On Approximating the Dynamic and Discrete Network Flow Problem
Bubai Manna, Bodhayan Roy, Vorapong Suppakitpaisarn
TL;DR
The paper studies the dynamic network flow with discrete, weighted groups (MS-DWSF) and proves APX-hardness for general graphs, while delivering a Polynomial-Time Approximation Scheme (PTAS) for a restricted path-graph instance with a single facility and uniform edge capacities. The approach maps the flow problem to a ready-time minsum bin packing variant, and uses an epsilon-based partitioning of items together with a dynamic programming formulation on a labeled graph to construct quasi-feasible packings. This yields concrete algorithmic guarantees for evacuation planning with grouped coordination and ready-time constraints, highlighting a boundary between intractable and tractable instances. The results motivate extensions to trees and multiple destinations and suggest avenues for refining approximation techniques under more complex network topologies.
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.