Abstract
When optimising under uncertainty, it is desirable that solutions are robust to unexpected disruptions and changes. A possible formalisation of robustness is given by super solutions. An assignment to a set of decision variables is an (a, b, c) super solution if any change involving at most a variables can be repaired by changing at most b other variables; the repair solution should have a cost of at most c units worse than a non-robust optimum. We propose a method exploiting Logic Based Benders Decomposition to find super solutions to an optimisation problem for generic disruptions. The master deals with the original problem, while subproblems try to find repair solutions for each possible disruption. As a case study, we consider the Kidney Exchange Problem, for which our method scales dramatically better on realistic instances than a reformulation-based approach from the literature.
This is a preview of subscription content, log in via an institution to check access.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Similar content being viewed by others
References
Abraham, D.J., Blum, A., Sandholm, T.: Clearing algorithms for barter exchange markets: enabling nationwide kidney exchanges. In: Proceedings 8th ACM Conference on Electronic Commerce (EC-2007), San Diego, California, USA, pp. 295–304, 11–15 June 2007. https://doi.org/10.1145/1250910.1250954
Alvelos, F., Klimentova, X., Rais, A., Viana, A.: A compact formulation for maximizing the expected number of transplants in kidney exchange programs. J. Phys.: Conf. Ser. 616, 012011 (2015)
Anderson, R., Ashlagi, I., Gamarnik, D., Roth, A.E.: Finding long chains in kidney exchange using the traveling salesman problem. Proc. Nat. Acad. Sci. 112(3), 663–668 (2015)
Ashlagi, I., Roth, A.E.: New challenges in multihospital kidney exchange. Am. Econ. Rev. 102(3), 354–359 (2012). https://doi.org/10.1257/aer.102.3.354
Bofill, M., Busquets, D., Muñoz, V., Villaret, M.: Reformulation based maxsat robustness. Constraints 18(2), 202–235 (2013). https://doi.org/10.1007/s10601-012-9130-2
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). https://doi.org/10.1016/j.ejor.2013.05.025
Demirović, E., Schwind, N., Okimoto, T., Inoue, K.: Recoverable team formation: building teams resilient to change. In: Proceedings of the 17th International Conference on Autonomous Agents and MultiAgent Systems, pp. 1362–1370. International Foundation for Autonomous Agents and Multiagent Systems (2018)
Derrien, A., Petit, T., Zampelli, S.: A declarative paradigm for robust cumulative scheduling. In: O’Sullivan, B. (ed.) CP 2014. LNCS, vol. 8656, pp. 298–306. Springer, Cham (2014). https://doi.org/10.1007/978-3-319-10428-7_23
Dickerson, J.P., Manlove, D.F., Plaut, B., Sandholm, T., Trimble, J.: Position-indexed formulations for kidney exchange. In: Proceedings of EC, pp. 25–42. ACM (2016)
Dickerson, J.P., Procaccia, A.D., Sandholm, T.: Failure-aware kidney exchange. In: Proceedings of the Fourteenth ACM Conference on Electronic Commerce, pp. 323–340. ACM (2013)
Dickerson, J.P., Procaccia, A.D., Sandholm, T.: Price of fairness in kidney exchange. In: Proceedings of the 2014 International Conference on Autonomous Agents and Multi-Agent Systems, pp. 1013–1020. International Foundation for Autonomous Agents and Multiagent Systems (2014)
Ginsberg, M.L., Parkes, A.J., Roy, A.: Supermodels and robustness. In: AAAI/IAAI, pp. 334–339 (1998)
Glorie, K.M., van de Klundert, J.J., Wagelmans, A.P.: Kidney exchange with long chains: an efficient pricing algorithm for clearing barter exchanges with branch-and-price. Manufact. Serv. Oper. Manage. 16(4), 498–512 (2014)
Hebrard, E., Hnich, B., Walsh, T.: Super CSPs. In: Proceedings of the CP-03 Workshop on “Handling Change and Uncertainty”, Cork, Ireland (2003)
Hebrard, E., Hnich, B., Walsh, T.: Super solutions in constraint programming. In: Régin, J.-C., Rueher, M. (eds.) CPAIOR 2004. LNCS, vol. 3011, pp. 157–172. Springer, Heidelberg (2004). https://doi.org/10.1007/978-3-540-24664-0_11
Holland, A., O’Sullivan, B.: Super solutions for combinatorial auctions. In: Faltings, B.V., Petcu, A., Fages, F., Rossi, F. (eds.) CSCLP 2004. LNCS (LNAI), vol. 3419, pp. 187–200. Springer, Heidelberg (2005). https://doi.org/10.1007/11402763_14
Holland, A., O’Sullivan, B.: Robust solutions for combinatorial auctions. In: Proceedings 6th ACM Conference on Electronic Commerce (EC-2005), Vancouver, BC, Canada, pp. 183–192, 5–8 June 2005. https://doi.org/10.1145/1064009.1064029
Holland, A., O’Sullivan, B.: Weighted super solutions for constraint programs. In: Proceedings, The Twentieth National Conference on Artificial Intelligence and the Seventeenth Innovative Applications of Artificial Intelligence Conference, Pittsburgh, Pennsylvania, USA, pp. 378–383, 9–13 July 2005
Holland, A., O’Sullivan, B.: Truthful risk-managed combinatorial auctions. In: Proceedings of the 20th International Joint Conference on Artificial Intelligence IJCAI 2007, pp. 1315–1320. Morgan Kaufmann Publishers Inc., San Francisco (2007). http://dl.acm.org/citation.cfm?id=1625275.1625488
Hooker, J.N., Ottosson, G.: Logic-based benders decomposition. Math. Programm. 96(1), 33–60 (2003)
Klimentova, X., Alvelos, F., Viana, A.: A new branch-and-price approach for the kidney exchange problem. In: Murgante, B., Misra, S., Rocha, A.M.A.C., Torre, C., Rocha, J.G., Falcão, M.I., Taniar, D., Apduhan, B.O., Gervasi, O. (eds.) ICCSA 2014. LNCS, vol. 8580, pp. 237–252. Springer, Cham (2014). https://doi.org/10.1007/978-3-319-09129-7_18
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)
Mak-Hau, V.: On the kidney exchange problem: cardinality constrained cycle and chain problems on directed graphs: a survey of integer programming approaches. J. Comb. Optim. 33, 1–25 (2015).https://doi.org/10.1007/s10878-015-9932-4
Manlove, D.F., O’Malley, G.: Paired and altruistic kidney donation in the UK: algorithms and experimentation. ACM J. Exp. Algorithmics 19(1) (2014). https://doi.org/10.1145/2670129
Mattei, N., Walsh, T.: PrefLib: a library for preferences http://www.preflib.org. In: Perny, P., Pirlot, M., Tsoukiàs, A. (eds.) ADT 2013. LNCS (LNAI), vol. 8176, pp. 259–270. Springer, Heidelberg (2013). https://doi.org/10.1007/978-3-642-41575-3_20
Pedroso, J.P.: Maximizing expectation on vertex-disjoint cycle packing. In: Murgante, B., et al. (eds.) ICCSA 2014. LNCS, vol. 8580, pp. 32–46. Springer, Cham (2014). https://doi.org/10.1007/978-3-319-09129-7_3
Plaut, B., Dickerson, J.P., Sandholm, T.: Fast optimal clearing of capped-chain barter exchanges. In: Proceedings of AAAI, pp. 601–607 (2016)
Powell, W.B.: A unified framework for optimization under uncertainty. In: Optimization Challenges in Complex, Networked and Risky Systems, pp. 45–83. INFORMS (2016)
Roy, A.: Fault tolerant boolean satisfiability. J. Artif. Intell. Res. 25, 503–527 (2006)
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)
Weigel, R., Bliek, C., Faltings, B.V.: On reformulation of constraint satisfaction problems (extended version). Artificial Intelligence, Citeseer (1998)
Acknowledgements
The Insight Centre for Data Analytics is supported by Science Foundation Ireland under Grant Number SFI/12/RC/2289, which is co-funded under the European Regional Development Fund.
Author information
Authors and Affiliations
Corresponding authors
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2019 Springer Nature Switzerland AG
About this paper
Cite this paper
Chisca, D.S., Lombardi, M., Milano, M., O’Sullivan, B. (2019). Logic-Based Benders Decomposition for Super Solutions: An Application to the Kidney Exchange Problem. In: Schiex, T., de Givry, S. (eds) Principles and Practice of Constraint Programming. CP 2019. Lecture Notes in Computer Science(), vol 11802. Springer, Cham. https://doi.org/10.1007/978-3-030-30048-7_7
Download citation
DOI: https://doi.org/10.1007/978-3-030-30048-7_7
Published:
Publisher Name: Springer, Cham
Print ISBN: 978-3-030-30047-0
Online ISBN: 978-3-030-30048-7
eBook Packages: Computer ScienceComputer Science (R0)