Stability in barter exchange markets

  • Sushmita GuptaEmail author
  • Fahad Panolan
  • Saket Saurabh
  • Meirav Zehavi


The notion of stability is the foundation of several classic problems in economics and computer science that arise in a wide-variety of real-world situations, including Stable Marriage, Stable Roommate, Hospital Resident and Group Activity Selection. We study this notion in the context of barter exchange markets. The input of our problem of interest consists of a set of people offering goods/services, with each person subjectively assigning values to a subset of goods/services offered by other people. The goal is to find a stable transaction, a set of cycles that is stable in the following sense: there does not exist a cycle such that every person participating in that cycle prefers to his current “status”. For example, consider a market where families are seeking vacation rentals and offering their own homes for the same. Each family wishes to acquire a vacation home in exchange of its own home without any monetary exchange. We study such a market by analyzing a stable transaction of houses involving cycles of fixed length. The underlying rationale is that an entire trade/exchange fails if any of the participating agents cancels the agreement; as a result, shorter (trading) cycles are desirable. We show that given a transaction, it can be verified whether or not it is stable in polynomial time, and that the problem of finding a stable transaction is NP-hard even if each person desires only a small number of other goods/services. Having established these results, we study the problem of finding a stable transaction in the framework of parameterized algorithms.


Algorithm design Stability Barter exchange FPT, W[1]-hard 



We thank anonymous reviewers for their valuable comments and suggestions which helped to improve the paper.


  1. 1.
    Abraham, D. J., Blum, A., & Sandholm, T. (2007). Clearing algorithms for barter exchange markets: Enabling nationwide kidney exchanges. In Proceedings of the 8th ACM conference on electronic commerce (EC) (pp. 295–304).Google Scholar
  2. 2.
    Anderson, R., Ashlagi, I., Gamarnik, D., & Roth, A. E. (2015). Finding long chains in kidney exchange using the traveling salesman problem. In Proceedings of the National Academy of Sciences (Vol. 112, pp. 663–668).Google Scholar
  3. 3.
    Biró, P. (2007). Stable exchange of indivisible goods with restrictions. In Proceedings of the 5th Japanese-Hungarian Symposium (pp. 97–105). Citeseer.Google Scholar
  4. 4.
    Brandt, F., Conitzer, V., Endrisss, U., Lang, J., & Procaccia, A. D. (Eds.). (2016). Handbook of computational social choice. Cambridge: Cambridge University Press.Google Scholar
  5. 5.
    Brandt, F., & Geist, C. (2016). Finding strategyproof social choice functions via SAT solving. Journal of Artificial Intelligence Research (JAIR), 55, 565–602.MathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    Cygan, M., Fomin, F. V., Kowalik, L., Lokshtanov, D., Marx, D., Pilipczuk, M., et al. (2015). Parameterized algorithms. Berlin: Springer.CrossRefzbMATHGoogle Scholar
  7. 7.
    Dickerson, J., Manlove, D., Plaut, B., Sandholm, T., & Trimble, J. (2016). Position-indexed formulations for kidney exchange. In Proceedings of EC.Google Scholar
  8. 8.
    Diestel, R. (2012). Graph theory (4th ed., Vol. 173)., Graduate texts in mathematics Berlin: Springer.zbMATHGoogle Scholar
  9. 9.
    Downey, R. G., & Fellows, M. R. (1995). Fixed-parameter tractability and completeness II: On completeness for W[1]. Theoretical Computer Science, 141(1&2), 109–131.MathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    Drummond, J., Perrault, A., & Bacchus, F. (2015). Sat is an effective and complete method for solving stable matching problems with couples. In Proceedings of IJCAI.Google Scholar
  11. 11.
    Fomin, F. V., Lokshtanov, D., Panolan, F., & Saurabh, S. (2016). Efficient computation of representative families with applications in parameterized and exact algorithms. Journal of the ACM, 63(4), 29:1–29:60.MathSciNetCrossRefzbMATHGoogle Scholar
  12. 12.
    Gale, D., & Shapley, L. S. (1962). College admissions and the stability of marriage. The American Mathematical Monthly, 69(1), 9–15.MathSciNetCrossRefzbMATHGoogle Scholar
  13. 13.
    Garey, M. R., & Johnson, D. S. (1979). Computers and intractability: A guide to the theory of NP-completeness. New York: W.H. Freeman.zbMATHGoogle Scholar
  14. 14.
    Gusfield, D., & Irving, R. W. (1989). The stable marriage problem: Structure and algorithms., Foundations of computing series Cambridge: MIT Press.zbMATHGoogle Scholar
  15. 15.
    Igarashi, A., Bredereck, R., & Elkind, E. (2017). On parameterized complexity of group activity selection problems on social networks. In Proceedings of AAMAS’17.Google Scholar
  16. 16.
    Igarashi, A., & Elkind, E. (2016). Hedonic games with graph-restricted communication. In Proceedings of AAMAS’16 (pp. 242–250).Google Scholar
  17. 17.
    Igarashi, A., Elkind, E., & Peters, D. (2017). Group activity selection on social network. In Proceedings of AAAI’17.Google Scholar
  18. 18.
  19. 19.
    Irving, R. W. (2007). The cycle roommates problem: A hard case of kidney exchange. Information Processing Letters, 103(1), 1–4.MathSciNetCrossRefzbMATHGoogle Scholar
  20. 20.
    Irving, R. W. (1994). Stable marriage and indifference. Discrete Applied Mathematics, 48(3), 261–272.MathSciNetCrossRefzbMATHGoogle Scholar
  21. 21.
    Irving, R., Iwama, K., Manlove, D. F., Miyazaki, S., & Morita, Y. (2002). Hard variants of stable marriage. Theoretical Computer Science, 276(1–2), 261–279.MathSciNetzbMATHGoogle Scholar
  22. 22.
    Knuth, D. E. (1997). Stable marriage and its relation to other combinatorial problems : An introduction to the mathematical analysis of algorithms. In CRM proceedings & lecture notes. Providence, RI: American Mathematical Society.Google Scholar
  23. 23.
    Manlove, D. F. (2013). Algorithmics of matching under preferences (Vol. 2)., Series on theoretical computer science Singapore: World Scientific.CrossRefzbMATHGoogle Scholar
  24. 24.
    McBride, I. (2015). Complexity results and integer programming models for hospitals/residents problem variants. Ph.D. thesis, University of Glasgow.Google Scholar
  25. 25.
  26. 26.
  27. 27.
    Sonmez, T. (1999). Strategy-proofness and essentially single-valued cores. Econometrica, 67(3), 677–689.MathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© Springer Science+Business Media, LLC, part of Springer Nature 2019

Authors and Affiliations

  1. 1.National Institute for Science Education and ResearchHBNIJatniIndia
  2. 2.Department of InformaticsUniversity of BergenBergenNorway
  3. 3.The Institute of Mathematical SciencesHBNIChennaiIndia
  4. 4.Ben-Gurion University of the NegevBeershebaIsrael

Personalised recommendations