Dantzig-Wolfe and Arc-Flow Reformulations: A Systematic Comparison

TL;DR

This paper systematically compares Dantzig-Wolfe and Arc-Flow reformulations, clarifying their theoretical connections and computational trade-offs, and provides practical guidelines for their use in large-scale integer optimization.

Contribution

It unifies the understanding of both reformulations, establishes conditions for translating valid inequalities, and empirically benchmarks their performance on vehicle routing problems.

Findings

Arc-Flow benefits from faster convergence in certain conditions

Dantzig-Wolfe is more efficient with a compact master problem

The two reformulations have comparable LP relaxations but differ in computational trade-offs

Abstract

Dantzig-Wolfe reformulation is a widely used technique for obtaining stronger relaxations in the context of branch-and-bound methods for solving integer optimization problems. Arc-Flow reformulations are a lesser known technique related to dynamic programming that has experienced a resurgence as result of the recent popularization of decision diagrams as a tool for solving integer programs. Although these two reformulation techniques arose independently, the recently proposed solution paradigm known as column elimination has revealed that they are in fact closely connected. Building on a unified formulation and notation, this study clarifies the theoretical connections and computational trade-offs between these two reformulations. We first revisit the known fact that the LP relaxations of these two reformulations yield the same dual bound. We then dig deeper, establishing conditions under which valid inequalities in the original, Dantzig-Wolfe, or Arc-Flow spaces can be translated across reformulations without loss of strength, and reinterpreting iterative strengthening methods, such as decremental state-space relaxation and column elimination, through the lens of cutting planes. To assess the potential impact of these insights empirically, we benchmark both reformulations under identical conditions on the vehicle routing problem with time windows using state-of-the-art column- and cut-generation techniques. The results identify clear contrasts: the Arc-Flow reformulation benefits from faster convergence and performs best when subproblems are highly relaxed or low-dimensional, whereas the Dantzig-Wolfe reformulation is more efficient when the master problem remains compact. Overall, our study provides a unified perspective and practical guidelines for choosing between Dantzig-Wolfe and Arc-Flow reformulations in large-scale integer optimization.