A robust optimization approach to flow decomposition

TL;DR

This work generalizes the minimum flow decomposition problem to inexact edge flows bounded by $f^l_{uv}$ and $f^u_{uv}$ and develops a robust optimization framework, including two adjustable formulations (MA and LA) for the weighted inexact case MWIFDP. It establishes the computational complexity landscape, showing polynomial-time solvability in key special cases and NP-hardness in general, and it introduces strict and adjustable robustness with budgeted and interval uncertainty sets. The paper further presents two adjustable robust problem formulations, along with a column-and-constraint generation approach, and demonstrates via computational experiments that adaptability substantially reduces the number of paths and improves robustness compared to solving each scenario independently. These findings highlight practical benefits for applications in genomics and transportation, and point to future opportunities in scalable solution methods and alternative uncertainty models.

Abstract

In this paper, we generalize the minimum flow decomposition problem (MFD) to incorporate uncertain edge capacities and tackle it 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. MFD problems permeate multiple important applications, including reconstructing genomic sequences to representing the flow of goods or passengers in distribution networks. Inspired by these applications, we generalize the MFD to an inexact case with bounded 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.

Figures (9)

• Figure 1: Example of a flow network and a flow decomposition into 5 $s$-$t$-paths dias2023accurate.
• Figure 2: Illustration of transformed graph $G'(V',E',d,c,\underaccent{\bar{}}{f},\bar{f})$ for the MCFP to solve the MWIFDP for $a_y = 0$ and $a_w > 0$ in $G(V,E,f^l,f^u)$.
• Figure 3: Graph structures of the original instances.
• Figure 4: Results of MA, LA and the naive approach for instance set small for the different uncertainty set sizes $|\mathcal{U}(\Gamma)|$ and values for $\Gamma'$.
• Figure 5: Objective values of MA, LA and the naive approach for the different uncertainty set sizes $|\mathcal{U}(\Gamma)|$ and values for $\Gamma'$ (logarithmic scale).

Theorems & Definitions (35)

• Definition 1: Flow network
• Definition 2: $k$-flow decomposition
• Remark 3
• Example 5: dias2023accurate
• Corollary 8
• Lemma 9
• proof
• Definition 10: Minimum-cost flow problem orlin1988faster
• Lemma 11
• proof