New and Simple Algorithms for Stable Flow Problems
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 e...
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Published in | Algorithmica Vol. 81; no. 6; pp. 2557 - 2591 |
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Main Authors | , |
Format | Journal Article |
Language | English |
Published |
New York
Springer US
01.06.2019
Springer Nature B.V |
Subjects | |
Online Access | Get full text |
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Summary: | 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
NP
-complete to decide whether an integral solution exists. |
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ISSN: | 0178-4617 1432-0541 |
DOI: | 10.1007/s00453-018-00544-7 |