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On the Computational Complexity of Non-dictatorial Aggregation

  • Lefteris Kirousis
  • Phokion G. Kolaitis
  • John LivieratosEmail author
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 11194)

Abstract

We investigate when non-dictatorial aggregation is possible from an algorithmic perspective, where non-dictatorial aggregation means that the votes cast by the members of a society can be aggregated in such a way that the collective outcome is not simply the choices made by a single member of the society. We consider the setting in which the members of a society take a position on a fixed collection of issues, where for each issue several different alternatives are possible, but the combination of choices must belong to a given set X of allowable voting patterns. Such a set X is called a possibility domain if there is an aggregator that is non-dictatorial, operates separately on each issue, and returns values among those cast by the society on each issue. We design a polynomial-time algorithm that decides, given a set X of voting patterns, whether or not X is a possibility domain. Furthermore, if X is a possibility domain, then the algorithm constructs in polynomial time such a non-dictatorial aggregator for X. We also design a polynomial-time algorithm that decides whether X is a uniform possibility domain, that is, whether X admits an aggregator that is non-dictatorial even when restricted to any two positions for each issue. As in the case of possibility domains, the algorithm also constructs in polynomial time a uniform non-dictatorial aggregator, if one exists.

Notes

Acknowledgements

The research of Lefteris Kirousis was partially supported by the Special Account for Research Grants of the National and Kapodistrian University of Athens. The work of Phokion G. Kolaitis is partially supported by NSF Grant IIS-1814152.

References

  1. 1.
    Arrow, K.J.: Social Choice and Individual Values. Wiley, New York (1951)Google Scholar
  2. 2.
    Bessiere, C., Carbonnel, C., Hebrard, E., Katsirelos, G., Walsh, T.: Detecting and exploiting subproblem tractability. In: IJCAI, pp. 468–474 (2013)Google Scholar
  3. 3.
    Bulatov, A.A.: A dichotomy theorem for constraint satisfaction problems on a 3-element set. J. ACM (JACM) 53(1), 66–120 (2006)MathSciNetCrossRefGoogle Scholar
  4. 4.
    Bulatov, A.A.: Complexity of conservative constraint satisfaction problems. ACM Trans. Comput. Logic (TOCL) 12(4), 24 (2011)MathSciNetCrossRefGoogle Scholar
  5. 5.
    Carbonnel, C.: The dichotomy for conservative constraint satisfaction is polynomially decidable. In: Rueher, M. (ed.) CP 2016. LNCS, vol. 9892, pp. 130–146. Springer, Cham (2016).  https://doi.org/10.1007/978-3-319-44953-1_9CrossRefGoogle Scholar
  6. 6.
    Carbonnel, C.: The meta-problem for conservative Mal’tsev constraints. In: Thirtieth AAAI Conference on Artificial Intelligence (AAAI-2016) (2016)Google Scholar
  7. 7.
    Dokow, E., Holzman, R.: Aggregation of binary evaluations. J. Econ. Theory 145(2), 495–511 (2010)MathSciNetCrossRefGoogle Scholar
  8. 8.
    Dokow, E., Holzman, R.: Aggregation of non-binary evaluations. Adv. Appl. Math. 45(4), 487–504 (2010)MathSciNetCrossRefGoogle Scholar
  9. 9.
    Endriss, U.: Judgment aggregation. In: Brandt, F., Conitzer, V., Endriss, U., Lang, J., Procaccia, A.D. (eds.), Handbook of Computational Social Choice, pp. 399–426. Cambridge University Press (2016)Google Scholar
  10. 10.
    Kirousis, L., Kolaitis, P.G., Livieratos, J.: Aggregation of votes with multiple positions on each issue. In: Höfner, P., Pous, D., Struth, G. (eds.) RAMICS 2017. LNCS, vol. 10226, pp. 209–225. Springer, Cham (2017).  https://doi.org/10.1007/978-3-319-57418-9_13CrossRefGoogle Scholar
  11. 11.
    Larose, B.: Algebra and the complexity of digraph CSPs: a survey. In: Dagstuhl Follow-Ups, vol. 7. Schloss Dagstuhl-Leibniz-Zentrum fuer Informatik (2017)Google Scholar
  12. 12.
    List, C., Puppe, C.: Judgment Aggregation: A Survey (2009)Google Scholar
  13. 13.
    Nehring, K., Puppe, C.: Strategy-proof social choice on single-peaked domains: possibility, impossibility and the space between. University of California at Davis (2002). http://vwl1.ets.kit.edu/puppe.php
  14. 14.
    Schaefer, T.J.: The complexity of satisfiability problems. In: Proceedings of the 10th Annual ACM Symposium on Theory of Computing, pp. 216–226 (1978)Google Scholar
  15. 15.
    Sharir, M.: A strong-connectivity algorithm and its applications in data flow analysis. Comput. Math. Appl. 7(1), 67–72 (1981)MathSciNetCrossRefGoogle Scholar
  16. 16.
    Szegedy, M., Xu, Y.: Impossibility theorems and the universal algebraic toolkit. CoRR, abs/1506.01315 (2015)Google Scholar
  17. 17.
    Szendrei, Á.: Clones in Universal Algebra, vol. 99. Presses de l’Université de Montréal, Montreal (1986)Google Scholar
  18. 18.
    Tarjan, R.E.: Depth-first search and linear graph algorithms. SIAM J. Comput. 1(2), 146–160 (1972)MathSciNetCrossRefGoogle Scholar

Copyright information

© Springer Nature Switzerland AG 2018

Authors and Affiliations

  1. 1.Department of MathematicsNational and Kapodistrian University of AthensAthensGreece
  2. 2.Computer Science DepartmentUC Santa Cruz and IBM Research - AlmadenSanta CruzUSA

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