A robust optimization approach to flow decomposition

Abstract: In this paper, we generalize the minimum flow decomposition problem (MFD) and incorporate uncertain edge capacities from the perspective of robust optimization. In the classical flow decomposition problem, a network flow is decomposed into a set of weighted paths from a fixed source node to a fixed sink node that precisely represents the flow distribution across all edges. While MFDs are often used in bioinformatics applications, they are also applicable in other fields, representing the flow of goods or passengers in distribution networks, where the decomposition represents the vehicles and corresponding capacities needed to cover these flows. Motivated by these applications, we generalize the MFD to the weighted inexact case with lower and upper bounds on the flow values, provide a detailed analysis, and explore different variants that are solvable in polynomial time. Moreover, we introduce the concept of robust flow decomposition by incorporating uncertain bounds and applying different robustness concepts to handle the uncertainty. Finally, we present two different adjustably robust problem formulations and perform computational experiments illustrating the benefit of adjustability.