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New and Simple Algorithms for Stable Flow Problems

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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.

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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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Authors and Affiliations

  1. Institute of Economics, Centre for Economic and Regional Studies, Hungarian Academy of Sciences, Tóth Kálmán u. 4., Budapest, 1097, Hungary

    Ágnes Cseh

  2. TUM School of Management and Department of Mathematics, Technische Universität München, Arcisstraße 21, 80333, Munich, Germany

    Jannik Matuschke

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  1. Ágnes Cseh
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Correspondence to Ágnes Cseh.

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A preliminary version of this paper appeared at the 43rd International Workshop on Graph-Theoretic Concepts in Computer Science (WG 2017). The authors were supported by Cooperation of Excellences Grant (KEP-6/2018), by the Ministry of Human Resources under its New National Excellence Programme (UNKP-18-4-BME-331), the Hungarian Academy of Sciences under its Momentum Programme (LP2016-3/2016), its János Bolyai Research Fellowship, OTKA Grant K128611, COST Action IC1205 on Computational Social Choice, and by the Alexander von Humboldt Foundation with funds of the German Federal Ministry of Education and Research (BMBF).

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Cseh, Á., Matuschke, J. New and Simple Algorithms for Stable Flow Problems. Algorithmica 81, 2557–2591 (2019). https://doi.org/10.1007/s00453-018-00544-7

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