Skip to main content
SpringerLink
Log in

New and Simple Algorithms for Stable Flow Problems

  • Published:
Algorithmica Aims and scope Submit manuscript

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.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Subscribe and save

Springer+ Basic
$34.99 /Month
  • Get 10 units per month
  • Download Article/Chapter or eBook
  • 1 Unit = 1 Article or 1 Chapter
  • Cancel anytime
Subscribe now

Buy Now

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Fig. 1
Fig. 2
Fig. 3
Fig. 4
Fig. 5
Fig. 6
Fig. 7
Fig. 8
Fig. 9
Fig. 10
Fig. 11

Similar content being viewed by others

References

  1. Baïou, M., Balinski, M.: Many-to-many matching: stable polyandrous polygamy (or polygamous polyandry). Discrete Appl. Math. 101, 1–12 (2000)

    Article  MathSciNet  MATH  Google Scholar 

  2. Balinski, M., Sönmez, T.: A tale of two mechanisms: student placement. J. Econ. Theory 84, 73–94 (1999)

    Article  MathSciNet  MATH  Google Scholar 

  3. Biró, P., Kern, W., Paulusma, D., Wojuteczky, P.: The stable fixtures problem with payments. Games Econ. Behav. 108, 245–268 (2017)

    Article  MathSciNet  MATH  Google Scholar 

  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. Chen, Y., Sönmez, T.: Improving efficiency of on-campus housing: an experimental study. Am. Econ. Rev. 92, 1669–1686 (2002)

    Article  Google Scholar 

  6. Cseh, Á., Manlove, D.F.: Stable marriage and roommates problems with restricted edges: complexity and approximability. Discrete Optim. 20, 62–89 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  7. Cseh, Á., Matuschke, J., Skutella, M.: Stable flows over time. Algorithms 6, 532–545 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  8. Dean, B.C., Munshi, S.: Faster algorithms for stable allocation problems. Algorithmica 58, 59–81 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  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)

    Article  MathSciNet  MATH  Google Scholar 

  10. Feder, T.: A new fixed point approach for stable networks and stable marriages. J. Comput. Syst. Sci. 45, 233–284 (1992)

    Article  MathSciNet  MATH  Google Scholar 

  11. Feder, T.: Network flow and 2-satisfiability. Algorithmica 11, 291–319 (1994)

    Article  MathSciNet  MATH  Google Scholar 

  12. Fleiner, T.: On the stable \(b\)-matching polytope. Math. Soc. Sci. 46, 149–158 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  13. Fleiner, T.: On stable matchings and flows. Algorithms 7, 1–14 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  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)

    Article  MATH  Google Scholar 

  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)

  16. Fleiner, T., Jankó, Z., Schlotter, I., Teytelboym, A.: Complexity of stability in trading networks. arXiv preprint arXiv:1805.08758 (2018)

  17. Ford, L.R., Fulkerson, D.R.: Flows in Networks. Princeton University Press, Princeton (1962)

    MATH  Google Scholar 

  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)

  19. Gale, D., Shapley, L.S.: College admissions and the stability of marriage. Am. Math. Mon. 69, 9–15 (1962)

    Article  MathSciNet  MATH  Google Scholar 

  20. Gale, D., Sotomayor, M.: Some remarks on the stable matching problem. Discrete Appl. Math. 11, 223–232 (1985)

    Article  MathSciNet  MATH  Google Scholar 

  21. Garey, M.R., Johnson, D.S.: Computers and Intractability. Freeman, San Francisco (1979)

    MATH  Google Scholar 

  22. Gusfield, D., Irving, R.W.: The Stable Marriage Problem: Structure and Algorithms. MIT Press, Cambridge (1989)

    MATH  Google Scholar 

  23. Irving, R.W., Leather, P., Gusfield, D.: An efficient algorithm for the “optimal” stable marriage. J. ACM 34, 532–543 (1987)

    Article  MathSciNet  Google Scholar 

  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)

  25. Jewell, W.S.: Multi-commodity Network Solutions. Operations Research Center, University of California, California (1966)

    MATH  Google Scholar 

  26. Király, T., Pap, J.: A note on kernels and Sperner’s Lemma. Discrete Appl. Math. 157, 3327–3331 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  27. Király, T., Pap, J.: Stable multicommodity flows. Algorithms 6, 161–168 (2013). https://doi.org/10.3390/a6010161

    Article  MathSciNet  MATH  Google Scholar 

  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)

  29. Lin, Y.S., Nguyen, T.: On variants of network flow stability. arXiv preprint arXiv:1710.03091 (2017)

  30. Ostrovsky, M.: Stability in supply chain networks. Am. Econ. Rev. 98, 897–923 (2008)

    Article  Google Scholar 

  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)

    Article  MathSciNet  MATH  Google Scholar 

  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)

    Article  MathSciNet  MATH  Google Scholar 

  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)

    Article  Google Scholar 

  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)

    Book  MATH  Google Scholar 

  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)

  36. Tardos, É.: A strongly polynomial algorithm to solve combinatorial linear programs. Oper. Res. 34, 250–256 (1986)

    Article  MathSciNet  MATH  Google Scholar 

Download references

Acknowledgements

We thank Tamás Fleiner for discussions on Lemma 3, and our reviewers for their suggestions that significantly improved the presentation of the paper.

Author information

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

Authors
  1. Ágnes Cseh
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Jannik Matuschke
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Ágnes Cseh.

Additional information

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

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).

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

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

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00453-018-00544-7

Keywords

Search

Navigation