Skip to main content

Linearized formulations for failure aware barter exchange


Mathematical programming formulations are developed for determining chains of organ-donation exchange pairs in a compatibility graph where pairwise exchanges may fail. The objective is to maximize the expected value where pairs are known to fail with given probabilities. In previous work, namely that of Dickerson et al. (Manag Sci 65(4):323–340, 2019) this NP-hard problem was solved heuristically or exactly only for limited path lengths. Although the problem appears highly nonlinear, we formulate it as a mixed-integer linear program. A computationally tractable layered formulation that approximately solves larger instances is also proposed and a computational study is presented for evaluating the proposed formulations.

This is a preview of subscription content, access via your institution.

Fig. 1
Fig. 2

Availability of data and material

The datasets generated during and/or analysed during the current study are available in the PSPP repository,


  1. Abraham, D.J., Blum, A., Sandholm, T.: Clearing algorithms for barter exchange markets: enabling nationwide kidney exchanges. In: MacKie-Mason, J.K., Parkes, D.C., Resnick, P. (eds.) Proceedings 8th ACM Conference on Electronic Commerce (EC-2007), San Diego, California, USA, June 11–15, 2007, pp. 295–304. ACM (2007)

  2. Anderson, R., Ashlagi, I., Gamarnik, D., Roth, A.E.: Finding long chains in kidney exchange using the traveling salesman problem. Proc. Natl. Acad. Sci. 112(3), 663–668 (2015)

    Article  Google Scholar 

  3. Ashlagi, I., Gamarnik, D., Rees, M.A., Roth, A.E.: The need for (long) chains in kidney exchange. Technical report, National Bureau of Economic Research (2012)

  4. Ashlagi, I., Gilchrist, D.S., Roth, A.E., Rees, M.A.: Nonsimultaneous chains and dominos in kidney-paired donation-revisited. Am. J. Transpl. 11, 984–994 (2011)

    Article  Google Scholar 

  5. Carvalho, M., Klimentova, X., Glorie, K., Viana, A., Constantino, M.: Robust models for the kidney exchange problem. INFORMS J. Comput. (2020).

  6. Chen, Y., Li, Y., Kalbfleisch, J.D., Zhou, Y., Leichtman, A., Song, P.X.-K.: Graph-based optimization algorithm and software on kidney exchanges. IEEE Trans. Biomed. Eng. 59(7), 1985–1991 (2012)

    Article  Google Scholar 

  7. Constantino, M., Klimentova, X., Viana, A., Rais, A.: New insights on integer-programming models for the kidney exchange problem. Eur. J. Oper. Res. 231(1), 57–68 (2013)

    Article  MathSciNet  Google Scholar 

  8. Dickerson, J.P., Procaccia, A.D., Sandholm, T.: Optimizing kidney exchange with transplant chains: theory and reality. In: Proceedings of the 11th International Conference on Autonomous Agents and Multiagent Systems, vol. 2, pp. 711–718 (2012)

  9. Dickerson, J.P., Procaccia, A.D., Sandholm, T.: Failure-aware kidney exchange. Manag. Sci. 65(4), 323–340 (2019)

    Article  Google Scholar 

  10. Dunning, I., Huchette, J., Lubin, M.: JuMP: a modeling language for mathematical optimization. SIAM Rev. 59(2), 295–320 (2017)

    Article  MathSciNet  Google Scholar 

  11. Garey, M.R., Johnson, D.S.: Computers and Intractability: A Guide to the Theory of NP-Completeness. W. H. Freeman, New York (1979)

    MATH  Google Scholar 

  12. Glorie, K.M., van de Klundert, J.J., Wagelmans, A.P.M.: Kidney exchange with long chains: an efficient pricing algorithm for clearing barter exchanges with branch-and-price. Manuf. Serv. Oper. Manag. 16, 498–512 (2014)

    Article  Google Scholar 

  13. Goldberg, N., Poss, M.: Maximum probabilistic all-or-nothing paths. Eur. J. Oper. Res. 283, 279–289 (2020)

    Article  MathSciNet  Google Scholar 

  14. Goldberg, N., Poss, M.: PSPP—GitHub repository. (2021)

  15. Goldberg, N., Rudolf, G.: On the complexity and approximation of the maximum expected value all-or-nothing subset. Discrete Appl. Math. 283, 1–10 (2020)

    Article  MathSciNet  Google Scholar 

  16. Gouveia, L.E.N., Leitner, M., Ruthmair, M.: Layered graph approaches for combinatorial optimization problems. Comput. Oper. Res. 102, 22–38 (2019)

    Article  MathSciNet  Google Scholar 

  17. Klimentova, X., Pedroso, J.P., Viana, A.: Maximising expectation of the number of transplants in kidney exchange programmes. Comput. Oper. Res. 73, 1–11 (2016)

    Article  MathSciNet  Google Scholar 

  18. Li, Y., Song, P.X.-K., Zhou, Y., Leichtman, A.B., Rees, M.A., Kalbfleisch, J.D.: Optimal decisions for organ exchanges in a kidney paired donation program. Stat. Biosci. 6, 85–104 (2014)

    Article  Google Scholar 

  19. Manlove, D.F., O’Malley, G.: Paired and altruistic kidney donation in the UK: algorithms and experimentation. ACM J. Exp. Algorithmics 19(1), 1–21 (2014)

    MathSciNet  Google Scholar 

  20. Mattei, N., Walsh, T.: Preflib: a library for preferences In: International Conference on Algorithmic Decision Theory, pp. 259–270. Springer (2013)

  21. McElfresh, D.C., Bidkhori, H., Dickerson, J.P.: Scalable robust kidney exchange. In: The Thirty-Third AAAI Conference on Artificial Intelligence, AAAI 2019, The Thirty-First Innovative Applications of Artificial Intelligence Conference, IAAI 2019, The Ninth AAAI Symposium on Educational Advances in Artificial Intelligence, EAAI 2019, Honolulu, Hawaii, USA, January 27–February 1, 2019, pp. 1077–1084. AAAI Press (2019)

  22. Rees, M.A., Kopke, J.E., Pelletier, R.P., Segev, D.L., Rutter, M.E., Fabrega, A.J., Rogers, J., Pankewycz, O.G., Hiller, J., Roth, A.E., Sandholm, T., Ünver, M.U., Montgomery, R.A.: A nonsimultaneous, extended, altruistic-donor chain. New Engl. J. Med. 360, 1096–1101 (2009)

    Article  Google Scholar 

  23. Roth, A.E., Sönmez, T., Ünver, M.U.: Efficient kidney exchange: Coincidence of wants in markets with compatibility-based preferences. Am. Econ. Rev. 97(3), 828–851 (2007)

    Article  Google Scholar 

  24. Saidman, S.L., Roth, A.E., Sönmez, T., Ünver, M.U., Delmonico, F.L.: Increasing the opportunity of live kidney donation by matching for two-and three-way exchanges. Transplantation 81(5), 773–782 (2006)

    Article  Google Scholar 

  25. Taccari, L.: Integer programming formulations for the elementary shortest path problem. Eur. J. Oper. Res. 252(1), 122–130 (2016)

    Article  MathSciNet  Google Scholar 

Download references

Author information

Authors and Affiliations


Corresponding author

Correspondence to Noam Goldberg.

Ethics declarations

Conflicts of interest/Competing interests


Code availability

Available upon request.

Ethics approval


Consent to participate


Consent for publication


Additional information

Publisher's Note

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

Rights and permissions

Reprints and Permissions

About this article

Verify currency and authenticity via CrossMark

Cite this article

Goldberg, N., Poss, M. Linearized formulations for failure aware barter exchange. Optim Lett 16, 1301–1313 (2022).

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI:


  • Longest path
  • MILP
  • Layered formulation
  • Clearing algorithms
  • Kidney exchange
  • Failure aware barter exchange