New and Simple Algorithms for Stable Flow Problems

• Published in: Algorithmica Vol. 81; no. 6; pp. 2557 - 2591
• Main Authors: Cseh, Ágnes, Matuschke, Jannik
• Format: Journal Article
• Language: English
• Published: New York Springer US 01.06.2019; Springer Nature B.V
• Subjects: Algorithm Analysis and Problem Complexity; Algorithms; Allocations; Apexes; Computer Science; Computer Systems Organization and Communication Networks; Data Structures and Information Theory; Graph theory; Mathematics of Computing; Polynomials; Theory of Computation; Restricted edges; Polynomial algorithm; NP-completeness; Stable flows; Multicommodity flows
• ISSN: 0178-4617; 1432-0541
• DOI: 10.1007/s00453-018-00544-7

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.