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Algorithmica

pp 1–35 | Cite as

New and Simple Algorithms for Stable Flow Problems

  • Ágnes CsehEmail author
  • Jannik Matuschke
Article
  • 24 Downloads

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.

References

  1. 1.
    Baïou, M., Balinski, M.: Many-to-many matching: stable polyandrous polygamy (or polygamous polyandry). Discrete Appl. Math. 101, 1–12 (2000)MathSciNetCrossRefzbMATHGoogle Scholar
  2. 2.
    Balinski, M., Sönmez, T.: A tale of two mechanisms: student placement. J. Econ. Theory 84, 73–94 (1999)MathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    Biró, P., Kern, W., Paulusma, D., Wojuteczky, P.: The stable fixtures problem with payments. Games Econ. Behav. 108, 245–268 (2017)MathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    Braun, S., Dwenger, N., Kübler, D.: Telling the truth may not pay off: an empirical study of centralized university admissions in Germany. B.E. J. Econ. Anal. Policy (2010).  https://doi.org/10.2202/1935-1682.2294
  5. 5.
    Chen, Y., Sönmez, T.: Improving efficiency of on-campus housing: an experimental study. Am. Econ. Rev. 92, 1669–1686 (2002)CrossRefGoogle Scholar
  6. 6.
    Cseh, Á., Manlove, D.F.: Stable marriage and roommates problems with restricted edges: complexity and approximability. Discrete Optim. 20, 62–89 (2016)MathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    Cseh, Á., Matuschke, J., Skutella, M.: Stable flows over time. Algorithms 6, 532–545 (2013)MathSciNetCrossRefGoogle Scholar
  8. 8.
    Dean, B.C., Munshi, S.: Faster algorithms for stable allocation problems. Algorithmica 58, 59–81 (2010)MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    Dias, V.M.F., da Fonseca, G.D., de Figueiredo, C.M.H., Szwarcfiter, J.L.: The stable marriage problem with restricted pairs. Theor. Comput. Sci. 306, 391–405 (2003)MathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    Feder, T.: A new fixed point approach for stable networks and stable marriages. J. Comput. Syst. Sci. 45, 233–284 (1992)MathSciNetCrossRefzbMATHGoogle Scholar
  11. 11.
    Feder, T.: Network flow and 2-satisfiability. Algorithmica 11, 291–319 (1994)MathSciNetCrossRefzbMATHGoogle Scholar
  12. 12.
    Fleiner, T.: On the stable \(b\)-matching polytope. Math. Soc. Sci. 46, 149–158 (2003)MathSciNetCrossRefzbMATHGoogle Scholar
  13. 13.
    Fleiner, T.: On stable matchings and flows. Algorithms 7, 1–14 (2014)MathSciNetCrossRefGoogle Scholar
  14. 14.
    Fleiner, T., Irving, R.W., Manlove, D.F.: Efficient algorithms for generalised stable marriage and roommates problems. Theor. Comput. Sci. 381, 162–176 (2007)CrossRefzbMATHGoogle Scholar
  15. 15.
    Fleiner, T., Jagadeesan, R., Jankó, Z., Teytelboym, A.: Trading networks with frictions. In: Proceedings of the 2018 ACM Conference on Economics and Computation. ACM, pp. 615–615 (2018)Google Scholar
  16. 16.
    Fleiner, T., Jankó, Z., Schlotter, I., Teytelboym, A.: Complexity of stability in trading networks. arXiv preprint arXiv:1805.08758 (2018)
  17. 17.
    Ford, L.R., Fulkerson, D.R.: Flows in Networks. Princeton University Press, Princeton (1962)zbMATHGoogle Scholar
  18. 18.
    Gai, A.T., Lebedev, D., Mathieu, F., de Montgolfier, F., Reynier, J., Viennot, L.: Acyclic preference systems in P2P networks. In: Kermarrec, A., Bougé, L., Priol, T. (eds.) Proceedings of Euro-Par ’07 (European Conference on Parallel and Distributed Computing): The 13th International Euro-Par Conference. Lecture Notes in Computer Science, vol. 4641, pp. 825–834. Springer (2007)Google Scholar
  19. 19.
    Gale, D., Shapley, L.S.: College admissions and the stability of marriage. Am. Math. Mon. 69, 9–15 (1962)MathSciNetCrossRefzbMATHGoogle Scholar
  20. 20.
    Gale, D., Sotomayor, M.: Some remarks on the stable matching problem. Discrete Appl. Math. 11, 223–232 (1985)MathSciNetCrossRefzbMATHGoogle Scholar
  21. 21.
    Garey, M.R., Johnson, D.S.: Computers and Intractability. Freeman, San Francisco (1979)zbMATHGoogle Scholar
  22. 22.
    Gusfield, D., Irving, R.W.: The Stable Marriage Problem: Structure and Algorithms. MIT Press, Cambridge (1989)zbMATHGoogle Scholar
  23. 23.
    Irving, R.W., Leather, P., Gusfield, D.: An efficient algorithm for the “optimal” stable marriage. J. ACM 34, 532–543 (1987)MathSciNetCrossRefGoogle Scholar
  24. 24.
    Jagadeesan, R.: Complementary inputs and the existence of stable outcomes in large trading networks. In: Proceedings of the 2017 ACM Conference on Economics and Computation. ACM, pp. 265–265 (2017)Google Scholar
  25. 25.
    Jewell, W.S.: Multi-commodity Network Solutions. Operations Research Center, University of California, California (1966)zbMATHGoogle Scholar
  26. 26.
    Király, T., Pap, J.: A note on kernels and Sperner’s Lemma. Discrete Appl. Math. 157, 3327–3331 (2009)MathSciNetCrossRefzbMATHGoogle Scholar
  27. 27.
    Király, T., Pap, J.: Stable multicommodity flows. Algorithms 6, 161–168 (2013).  https://doi.org/10.3390/a6010161 MathSciNetCrossRefGoogle Scholar
  28. 28.
    Knuth, D.: Mariages Stables. Les Presses de L’Université de Montréal (1976). In: English translation in Stable Marriage and its Relation to Other Combinatorial Problems, vol. 10 of CRM Proceedings and Lecture Notes, American Mathematical Society (1997)Google Scholar
  29. 29.
    Lin, Y.S., Nguyen, T.: On variants of network flow stability. arXiv preprint arXiv:1710.03091 (2017)
  30. 30.
    Ostrovsky, M.: Stability in supply chain networks. Am. Econ. Rev. 98, 897–923 (2008)CrossRefGoogle Scholar
  31. 31.
    Papadimitriou, C.H.: On the complexity of the parity argument and other inefficient proofs of existence. J. Comput. Syst. Sci. 48, 498–532 (1994)MathSciNetCrossRefzbMATHGoogle Scholar
  32. 32.
    Perach, N., Polak, J., Rothblum, U.G.: A stable matching model with an entrance criterion applied to the assignment of students to dormitories at the Technion. Int. Jo. Game Theory 36, 519–535 (2008)MathSciNetCrossRefzbMATHGoogle Scholar
  33. 33.
    Roth, A.E.: The evolution of the labor market for medical interns and residents: a case study in game theory. J. Polit. Econ. 92, 991–1016 (1984)CrossRefGoogle Scholar
  34. 34.
    Roth, A.E., Sotomayor, M.A.O.: Two-Sided Matching: A Study in Game-Theoretic Modeling and Analysis. Econometric Society Monographs, vol. 18. Cambridge University Press, Cambridge (1990)CrossRefzbMATHGoogle Scholar
  35. 35.
    Shepherd, F.B., Vetta, A., Wilfong, G.T.: Polylogarithmic approximations for the capacitated single-sink confluent flow problem. In: 2015 IEEE 56th Annual Symposium on Foundations of Computer Science (FOCS). IEEE, pp. 748–758 (2015)Google Scholar
  36. 36.
    Tardos, É.: A strongly polynomial algorithm to solve combinatorial linear programs. Oper. Res. 34, 250–256 (1986)MathSciNetCrossRefzbMATHGoogle Scholar

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Authors and Affiliations

  1. 1.Institute of Economics, Centre for Economic and Regional StudiesHungarian Academy of SciencesBudapestHungary
  2. 2.TUM School of Management and Department of MathematicsTechnische Universität MünchenMunichGermany

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