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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.

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References

  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)

  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)

    Article  MATH  Google Scholar 

  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)

    Article  MathSciNet  MATH  Google Scholar 

  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)

  5. Chassein, A., Goerigk, M.: Alternative formulations for the ordered weighted averaging objective. Inf. Process. Lett. 115(6), 604–608 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  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)

    MATH  Google Scholar 

  7. Chong, K.: An induction theorem for rearrangements. Cand. J. Math. 28, 154–160 (1976)

    Article  MathSciNet  MATH  Google Scholar 

  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)

  9. Edmonds, J., Karp, R.: Theoretical improvements in algorithmic efficiency for network flow problems. J. ACM 19(2), 248–264 (1972)

    Article  MATH  Google Scholar 

  10. Garey, M., Johnson, D.: Computers and Intractability. W.H. Freeman and company, Stuttgart (1979)

    MATH  Google Scholar 

  11. Garfinkel, R.: An improved algorithm for the bottleneck assignment problem. Oper. Res. 19(7), 1747–1751 (1971)

    Article  MATH  Google Scholar 

  12. Garg, N., Kavitha, T., Kumar, A., Mehlhorn, K., Mestre, J.: Assigning papers to referees. Algorithmica 58, 119–136 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  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)

  14. Gorski, J., Ruzika, S.: On k-max-optimization. Oper. Res. Lett. 37(1), 23–26 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  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)

  16. Gupta, S., Punnen, A.: Minimum deviation problems. Oper. Res. Lett. 7(4), 201–204 (1988)

    Article  MathSciNet  MATH  Google Scholar 

  17. Gupta, S., Punnen, A.: k-Sum optimization problems. Oper. Res. Lett. 9(2), 121–126 (1990)

    Article  MathSciNet  MATH  Google Scholar 

  18. Hardy, G., Littlewood, J.E., Pólya, G.: Inequalities. Cambridge University Press, Cambridge (1934)

    MATH  Google Scholar 

  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)

  20. Kalcsics, J., Nickel, S., Puerto, P.A.T.: Algorithmic results for ordered median problems. Oper. Res. Lett. 30(3), 149–158 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  21. Kuhn, H.: The Hungarian method for the assignment problem. Nav. Res. Logist. Q. 2(1–2), 83–97 (1955)

    Article  MathSciNet  MATH  Google Scholar 

  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)

    Article  MathSciNet  MATH  Google Scholar 

  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)

  24. Marshall, W., Olkin, I.: Inequalities: Theory of Majorization and its Applications. Academic Press, London (1979)

    MATH  Google Scholar 

  25. Minoux, M.: Solving combinatorial problems with combined min-max-min-sum objective and applications. Math. Program. 45(1–3), 361–372 (1989)

    Article  MathSciNet  MATH  Google Scholar 

  26. Moulin, H.: Axioms of Cooperative Decision Making. Monograph of the Econometric Society. Cambridge University Press, Cambridge (1988)

    Book  MATH  Google Scholar 

  27. Ogryczak, W.: Multicriteria models for fair resource allocation. Control Cybern. 36, 303–332 (2007)

    MathSciNet  MATH  Google Scholar 

  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

    MathSciNet  Google Scholar 

  29. Ogryczak, W., Sliwinski, T.: On solving linear programs with the ordered weighted averaging objective. Eur. J. Operat. Res. 148(1), 80–91 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  30. Schmeidler, D.: Integral representation without additivity. Proc. Am. Math. Soc. 97(2), 255–261 (1986)

    Article  MathSciNet  MATH  Google Scholar 

  31. Sen, A.: On Economic Inequality. Oxford University Press, Oxford (1973)

    Book  Google Scholar 

  32. Shorrocks, A.: Ranking income distributions. Economica 50, 3–17 (1983)

    Article  Google Scholar 

  33. Sokkalingam, P., Aneja, Y.: Lexicographic bottleneck combinatorial problems. Oper. Res. Lett. 23(1–2), 27–33 (1998)

    Article  MathSciNet  MATH  Google Scholar 

  34. Torra, V.: The weighted owa operator. Int. J. Intel. Syst. 12, 153–166 (1997)

    Article  MATH  Google Scholar 

  35. Weymark, J.: Generalized Gini inequality indices. Math. Soc. Sci. 1, 409–430 (1981)

    Article  MathSciNet  MATH  Google Scholar 

  36. Yager, R.: On ordered weighted averaging aggregation operators in multicriteria decision making. IEEE Trans. Syst Man Cybern. 18, 183–190 (1988)

    Article  MATH  Google Scholar 

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Acknowledgements

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

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Authors and Affiliations

  1. CNRS, LAMSADE, Université Paris-Dauphine, PSL Research University, 75016, Paris, France

    Julien Lesca

  2. CNRS, Laboratoire d’informatique de Paris 6, LIP6, Sorbonne Université, 75005, Paris, France

    Michel Minoux & Patrice Perny

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  1. Julien Lesca
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  2. Michel Minoux
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  3. Patrice Perny
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Correspondence to Julien Lesca.

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Lesca, J., Minoux, M. & Perny, P. The Fair OWA One-to-One Assignment Problem: NP-Hardness and Polynomial Time Special Cases. Algorithmica 81, 98–123 (2019). https://doi.org/10.1007/s00453-018-0434-5

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