Algorithmica

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The Fair OWA One-to-One Assignment Problem: NP-Hardness and Polynomial Time Special Cases

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Abstract

We consider a one-to-one assignment problem consisting of matching n objects with n agents. Any matching leads to a utility vector whose n components measure the satisfaction of the various agents. We want to find an assignment maximizing a global utility defined as an ordered weighted average (OWA) of the n individual utilities. OWA weights are assumed to be non-increasing with ranks of satisfaction so as to include an idea of fairness in the definition of social utility. We first prove that the problem is NP-hard; then we propose a polynomial algorithm under some restrictions on the set of admissible weight vectors, proving that the problem belongs to XP.

Keywords

Assignment problem Fairness Ordered weighted average Complexity 

Notes

Acknowledgements

The referees are gratefully acknowledged for their constructive comments and suggestions which resulted in an improved presentation of the paper.

References

  1. 1.
    Aziz, H., Gaspers, S., Mackenzie, S., Walsh, T.: Fair assignment of indivisible objects under ordinal preferences. In: Proceedings of the 13th International Conference on Autonomous Agents and Multiagent Systems, pp. 1305–1312 (2014)Google Scholar
  2. 2.
    Boland, N., Domínguez-Marín, P., Nickel, S., Puerto, J.: Exact procedures for solving the discrete ordered median problem. Comput. OR 33(11), 3270–3300 (2006)CrossRefMATHGoogle Scholar
  3. 3.
    Bouveret, S., Lang, J.: Efficiency and envy-freeness in fair division of indivisible goods: logical representation and complexity. J. Artif. Intell. Res. 32, 525–564 (2008)MathSciNetMATHGoogle Scholar
  4. 4.
    Bouveret, S., Lemaître, M., Fargier, H., Lang, J.: Allocation of indivisible goods: a general model and some complexity results. In: Proceedings of the 4th International Joint Conference on Autonomous Agents and Multiagent Systems, pp. 1309–1310 (2005)Google Scholar
  5. 5.
    Chassein, A., Goerigk, M.: Alternative formulations for the ordered weighted averaging objective. Inf. Process. Lett. 115(6), 604–608 (2015)MathSciNetCrossRefMATHGoogle Scholar
  6. 6.
    Chevaleyre, Y., Dunne, P., Endriss, U., Lang, J., Lemaître, M., Maudet, N., Padget, J., Phelps, S., Rodríguez-Aguilar, J., Sousa, P.: Issues in multiagent resource allocation. Informatica (Slovenia) 30(1), 3–31 (2006)MATHGoogle Scholar
  7. 7.
    Chong, K.: An induction theorem for rearrangements. Cand. J. Math. 28, 154–160 (1976)MathSciNetCrossRefMATHGoogle Scholar
  8. 8.
    Cook, S.A.: The complexity of theorem-proving procedures. In: Proceedings of the 3rd Annual ACM Symposium on Theory of Computing, pp. 151–158. ACM (1971)Google Scholar
  9. 9.
    Edmonds, J., Karp, R.: Theoretical improvements in algorithmic efficiency for network flow problems. J. ACM 19(2), 248–264 (1972)CrossRefMATHGoogle Scholar
  10. 10.
    Garey, M., Johnson, D.: Computers and Intractability. W.H. Freeman and company, Stuttgart (1979)MATHGoogle Scholar
  11. 11.
    Garfinkel, R.: An improved algorithm for the bottleneck assignment problem. Oper. Res. 19(7), 1747–1751 (1971)CrossRefMATHGoogle Scholar
  12. 12.
    Garg, N., Kavitha, T., Kumar, A., Mehlhorn, K., Mestre, J.: Assigning papers to referees. Algorithmica 58, 119–136 (2010)MathSciNetCrossRefMATHGoogle Scholar
  13. 13.
    Golden, B., Perny, P.: Infinite order Lorenz dominance for fair multiagent optimization. In: Proceedings of the 9th International Conference on Autonomous Agents and Multiagent Systems, pp. 383–390 (2010)Google Scholar
  14. 14.
    Gorski, J., Ruzika, S.: On k-max-optimization. Oper. Res. Lett. 37(1), 23–26 (2009)MathSciNetCrossRefMATHGoogle Scholar
  15. 15.
    Gourvès, L., Monnot, J., Tlilane, L.: A matroid approach to the worst case allocation of indivisible goods. In: Proceedings of the 23rd International Joint Conference on Artificial Intelligence, pp. 136–142 (2013)Google Scholar
  16. 16.
    Gupta, S., Punnen, A.: Minimum deviation problems. Oper. Res. Lett. 7(4), 201–204 (1988)MathSciNetCrossRefMATHGoogle Scholar
  17. 17.
    Gupta, S., Punnen, A.: k-Sum optimization problems. Oper. Res. Lett. 9(2), 121–126 (1990)MathSciNetCrossRefMATHGoogle Scholar
  18. 18.
    Hardy, G., Littlewood, J.E., Pólya, G.: Inequalities. Cambridge University Press, Cambridge (1934)MATHGoogle Scholar
  19. 19.
    Heinen, T., Nguyen, N.T., Rothe, J.: Fairness and rank-weighted utilitarianism in resource allocation. In: Proceedings of the 4th International Conference on Algorithmic Decision Theory, pp. 521–536 (2015)Google Scholar
  20. 20.
    Kalcsics, J., Nickel, S., Puerto, P.A.T.: Algorithmic results for ordered median problems. Oper. Res. Lett. 30(3), 149–158 (2002)MathSciNetCrossRefMATHGoogle Scholar
  21. 21.
    Kuhn, H.: The Hungarian method for the assignment problem. Nav. Res. Logist. Q. 2(1–2), 83–97 (1955)MathSciNetCrossRefMATHGoogle Scholar
  22. 22.
    Lesca, J., Minoux, M., Perny, P.: Compact versus noncompact LP formulations for minimizing convex Choquet integrals. Discrete Appl. Math. 161(1–2), 184–199 (2013)MathSciNetCrossRefMATHGoogle Scholar
  23. 23.
    Lesca, J., Perny, P.: LP Solvable Models for Multiagent Fair Allocation problems. In: Proceedings of the 19th European Conference on Artificial Intelligence, pp. 387–392 (2010)Google Scholar
  24. 24.
    Marshall, W., Olkin, I.: Inequalities: Theory of Majorization and its Applications. Academic Press, London (1979)MATHGoogle Scholar
  25. 25.
    Minoux, M.: Solving combinatorial problems with combined min-max-min-sum objective and applications. Math. Program. 45(1–3), 361–372 (1989)MathSciNetCrossRefMATHGoogle Scholar
  26. 26.
    Moulin, H.: Axioms of Cooperative Decision Making. Monograph of the Econometric Society. Cambridge University Press, Cambridge (1988)CrossRefMATHGoogle Scholar
  27. 27.
    Ogryczak, W.: Multicriteria models for fair resource allocation. Control Cybern. 36, 303–332 (2007)MathSciNetMATHGoogle Scholar
  28. 28.
    Ogryczak, W., Luss, H., Pióro, M., Nace, D., Tomaszewski, A.: Fair optimization and networks: a survey. J Appl. Math. 2014, 612018 (2014).  https://doi.org/10.1155/2014/612018 MathSciNetGoogle Scholar
  29. 29.
    Ogryczak, W., Sliwinski, T.: On solving linear programs with the ordered weighted averaging objective. Eur. J. Operat. Res. 148(1), 80–91 (2003)MathSciNetCrossRefMATHGoogle Scholar
  30. 30.
    Schmeidler, D.: Integral representation without additivity. Proc. Am. Math. Soc. 97(2), 255–261 (1986)MathSciNetCrossRefMATHGoogle Scholar
  31. 31.
    Sen, A.: On Economic Inequality. Oxford University Press, Oxford (1973)CrossRefGoogle Scholar
  32. 32.
    Shorrocks, A.: Ranking income distributions. Economica 50, 3–17 (1983)CrossRefGoogle Scholar
  33. 33.
    Sokkalingam, P., Aneja, Y.: Lexicographic bottleneck combinatorial problems. Oper. Res. Lett. 23(1–2), 27–33 (1998)MathSciNetCrossRefMATHGoogle Scholar
  34. 34.
    Torra, V.: The weighted owa operator. Int. J. Intel. Syst. 12, 153–166 (1997)CrossRefMATHGoogle Scholar
  35. 35.
    Weymark, J.: Generalized Gini inequality indices. Math. Soc. Sci. 1, 409–430 (1981)MathSciNetCrossRefMATHGoogle Scholar
  36. 36.
    Yager, R.: On ordered weighted averaging aggregation operators in multicriteria decision making. IEEE Trans. Syst Man Cybern. 18, 183–190 (1988)CrossRefMATHGoogle Scholar

Copyright information

© Springer Science+Business Media, LLC, part of Springer Nature 2018

Authors and Affiliations

  1. 1.CNRS, LAMSADEUniversité Paris-Dauphine, PSL Research UniversityParisFrance
  2. 2.CNRS, Laboratoire d’informatique de Paris 6, LIP6Sorbonne UniversitéParisFrance

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