Solving Unsplittable Network Flow Problems with Decision Diagrams

TL;DR

This paper introduces a decision diagram-based approach to efficiently solve unsplittable network flow problems with no-split no-merge constraints, especially in the context of stochastic unit train scheduling with uncertain demand.

Contribution

The paper presents a novel DD-based decomposition framework that reduces computational complexity for unsplittable network flow problems with stochastic demand, outperforming standard methods.

Findings

Significant reduction in solution time using the DD framework.

Effective handling of stochastic demand in train scheduling.

Improved scalability over traditional mixed-integer programming approaches.

Abstract

In unsplittable network flow problems, certain nodes must satisfy a combinatorial requirement that the incoming arc flows cannot be split or merged when routed through outgoing arcs. This so-called "no-split no-merge" requirement arises in unit train scheduling where train consists should remain intact at stations that lack necessary equipment and manpower to attach/detach them. Solving the unsplittable network flow problems with standard mixed-integer programming formulations is computationally difficult due to the large number of binary variables needed to determine matching pairs between incoming and outgoing arcs of nodes with no-split no-merge constraint. In this paper, we study a stochastic variant of the unit train scheduling problem where the demand is uncertain. We develop a novel decision diagram (DD)-based framework that decomposes the underlying two-stage formulation into a master problem that contains the combinatorial requirements, and a subproblem that models a continuous network flow problem. The master problem is modeled by a DD in a transformed space of variables with a smaller dimension, leading to a substantial improvement in solution time. Similarly to the Benders decomposition technique, the subproblems output cutting planes that are used to refine the master DD. Computational experiments show a significant improvement in solution time of the DD framework compared with that of standard methods.