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New and Simple Algorithms for Stable Flow Problems

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Graph-Theoretic Concepts in Computer Science (WG 2017)

Part of the book series: Lecture Notes in Computer Science ((LNTCS,volume 10520))

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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 [13] 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 allocation 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 highly complex stable multicommodity flow model by Király and Pap [24]. We present several graph-based reductions that show equivalence to a significantly simpler model. We further show that it is NP-complete to decide whether an integral solution exists.

The authors were supported by the Hungarian Academy of Sciences under its Momentum Programme (LP2016-3/2016) and its János Bolyai Research Scholarship, OTKA grant K108383 and 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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Acknowledgment

We thank Tamás Fleiner for discussions on Lemma 1.

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Authors and Affiliations

  1. Institute of Economics, Hungarian Academy of Sciences and Corvinus University of Budapest, Budaörsi út 45., Budapest, 1112, Hungary

    Ágnes Cseh

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

    Jannik Matuschke

Authors
  1. Ágnes Cseh
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  2. Jannik Matuschke
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Correspondence to Ágnes Cseh .

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Editors and Affiliations

  1. Utrecht University, Utrecht, The Netherlands

    Hans L. Bodlaender

  2. Rheinisch-Westfälische Technische Hochschule Aachen, Aachen, Germany

    Gerhard J. Woeginger

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Cseh, Á., Matuschke, J. (2017). New and Simple Algorithms for Stable Flow Problems. In: Bodlaender, H., Woeginger, G. (eds) Graph-Theoretic Concepts in Computer Science. WG 2017. Lecture Notes in Computer Science(), vol 10520. Springer, Cham. https://doi.org/10.1007/978-3-319-68705-6_16

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  • DOI: https://doi.org/10.1007/978-3-319-68705-6_16

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  • Publisher Name: Springer, Cham

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  • Online ISBN: 978-3-319-68705-6

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