Algorithmica

pp 1–35

# New and Simple Algorithms for Stable Flow Problems

• Ágnes Cseh
• Jannik Matuschke
Article

## Abstract

Stable flows generalize the well-known concept of stable matchings to markets in which transactions may involve several agents, forwarding flow from one to another. An instance of the problem consists of a capacitated directed network in which vertices express their preferences over their incident edges. A network flow is stable if there is no group of vertices that all could benefit from rerouting the flow along a walk. Fleiner (Algorithms 7:1–14, 2014) established that a stable flow always exists by reducing it to the stable allocation problem. We present an augmenting path algorithm for computing a stable flow, the first algorithm that achieves polynomial running time for this problem without using stable allocations as a black-box subroutine. We further consider the problem of finding a stable flow such that the flow value on every edge is within a given interval. For this problem, we present an elegant graph transformation and based on this, we devise a simple and fast algorithm, which also can be used to find a solution to the stable marriage problem with forced and forbidden edges. Finally, we study the stable multicommodity flow model introduced by Király and Pap (Algorithms 6:161–168, 2013). The original model is highly involved and allows for commodity-dependent preference lists at the vertices and commodity-specific edge capacities. We present several graph-based reductions that show equivalence to a significantly simpler model. We further show that it is $${\textsf {NP}}$$-complete to decide whether an integral solution exists.

## Keywords

Stable flows Restricted edges Multicommodity flows Polynomial algorithm NP-completeness

## Notes

### Acknowledgements

We thank Tamás Fleiner for discussions on Lemma 3, and our reviewers for their suggestions that significantly improved the presentation of the paper.

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