Autonomous Agents and Multi-Agent Systems

, Volume 32, Issue 5, pp 672–692 | Cite as

How hard is it to control a group?

  • Yongjie YangEmail author
  • Dinko Dimitrov


We consider group identification models in which the aggregation of individual opinions concerning who is qualified in a given society determines the set of socially qualified persons. In this setting, we study the extent to which social qualification can be changed when societies expand, shrink, or partition themselves. The answers we provide are with respect to the computational complexity of the corresponding control problems and fully cover the class of consent aggregation rules introduced by Samet and Schmeidler (J Econ Theory, 110(2):213–233, 2003) as well as procedural rules for group identification. We obtain both polynomial-time solvability results and NP-hardness results. In addition, we also study these problems from the parameterized complexity perspective, and obtain some fixed-parameter tractability results.


Group identification Consent rules Procedural rules Computational complexity Parameterized complexity Control 



We thank Shao-Chin Sung and the anonymous reviewers of JAAMAS and COMSOC 2016 for their valuable comments.


  1. 1.
    Aziz, H., Gaspers, S., Gudmundsson, J., Mackenzie, S., Mattei, N., & Walsh, T. (2015). Computational aspects of multi-winner approval voting. In AAMAS (pp. 107–115).Google Scholar
  2. 2.
    Bartholdi, J. J., III, Tovey, C. A., & Trick, M. A. (1992). How hard is it to control an election? Mathematical and Computer Modelling, 16(8–9), 27–40.Google Scholar
  3. 3.
    Baumeister, D., Erdélyi, G., Hemaspaandra, E., Hemaspaandra, L. A., & Rothe, J. (2010). Computational aspects of approval voting, chap. 10. In Handbook on approval voting (pp. 199–251). Berlin: Springer.Google Scholar
  4. 4.
    Berga, D., Bergantiños, G., Massó, J., & Neme, A. (2004). Stability and voting by committees with exit. Social Choice and Welfare, 23(2), 229–247.MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    Cho, W. J., & Ju, B. G. (2017). Multinary group identification. Theoretical Economics, 12(2), 513–531.MathSciNetCrossRefGoogle Scholar
  6. 6.
    Cook, S. A. (1971). The complexity of theorem-proving procedures. In STOC (pp. 151–158).Google Scholar
  7. 7.
    Cygan, M., Fomin, F. V., Kowalik, L., Lokshtanov, D., Marx, D., Pilipczuk, M., et al. (2015). Parameterized algorithms. Berlin: Springer.CrossRefzbMATHGoogle Scholar
  8. 8.
    Dimitrov, D. (2011). The social choice approach to group identification. In Consensual processes (pp. 123–134).Google Scholar
  9. 9.
    Dimitrov, D., Sung, S. C., & Xu, Y. (2007). Procedural group identification. Mathematical Social Sciences, 54(2), 137–146.MathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    Downey, R. G., & Fellows, M. R. (1999). Parameterized complexity. Berlin: Springer.CrossRefzbMATHGoogle Scholar
  11. 11.
    Downey, R. G., & Fellows, M. R. (2013). Fundamentals of parameterized complexity. Texts in computer science. Berlin: Springer.CrossRefzbMATHGoogle Scholar
  12. 12.
    Elkind, E., & Lackner, M. (2015) Structure in dichotomous preferences. In IJCAI (pp. 2019–2025).Google Scholar
  13. 13.
    Erdélyi, G., Reger, C., & Yang, Y. (2017). The complexity of bribery and control in group identification. In AAMAS (pp. 1142–1150).Google Scholar
  14. 14.
    Erdélyi, G., Reger, C., & Yang, Y. (2017). Complexity of group identification with partial information. In ADT (pp. 182–196).Google Scholar
  15. 15.
    Faliszewski, P., Hemaspaandra, E., Hemaspaandra, L. A., & Rothe, J. (2009). Llull and Copeland voting computationally resist bribery and constructive control. Journal of Artificial Intelligence Research, 35, 275–341.MathSciNetCrossRefzbMATHGoogle Scholar
  16. 16.
    Faliszewski, P., & Rothe, J. (2016). Control and bribery in voting. In F. Brandt, V. Conitzer, U. Endriss, J. Lang, & A. Procaccia (Eds.), Handbook of computational social choice (Chap. 7, pp. 146–168). Cambridge: Cambridge University Press.Google Scholar
  17. 17.
    Faliszewski, P., Slinko, A., & Talmon, N. (2017). The complexity of multiwinner voting rules with variable number of winners. arXiv:1711.06641
  18. 18.
    Fishburn, P. C., & Brams, S. J. (1981). Approval voting, Condorcet’s principle, and runoff elections. Public Choice, 36(1), 89–114.CrossRefGoogle Scholar
  19. 19.
    Frank, A., & Tardos, É. (1987). An application of simultaneous Diophantine approximaiton in combinatorial optimazation. Combinatorica, 7(1), 49–65.MathSciNetCrossRefzbMATHGoogle Scholar
  20. 20.
    Garey, M. R., & Johnson, D. S. (1979). Computers and intractability: A guide to the theory of NP-completeness. New York: W. H. Freeman.zbMATHGoogle Scholar
  21. 21.
    Gonzalez, T. F. (1985). Clustering to minimize the maximum intercluster distance. Theoretical Computer Science, 38, 293–306.MathSciNetCrossRefzbMATHGoogle Scholar
  22. 22.
    Hemaspaandra, E., Hemaspaandra, L. A., & Rothe, J. (2007). Anyone but him: The complexity of precluding an alternative. Artificial Intelligence, 171(5–6), 255–285.MathSciNetCrossRefzbMATHGoogle Scholar
  23. 23.
    Houy, N. (2007). “I want to be a J!”: Liberalism in group identification problems. Mathematical Social Sciences, 54(1), 59–70.MathSciNetCrossRefzbMATHGoogle Scholar
  24. 24.
    Kannan, R. (1987). Minkowski’s convex body theorem and integer programming. Mathematics of Operations Research, 12(3), 415–440.MathSciNetCrossRefzbMATHGoogle Scholar
  25. 25.
    Kasher, A., & Rubinstein, A. (1997). On the question “Who is a J?” A social choice approach. Logique & Analyse, 40(160), 385–395.MathSciNetzbMATHGoogle Scholar
  26. 26.
    Kilgour, D. M. (2016). Approval elections with a variable number of winners. Theory and Decision, 81(2), 199–211.MathSciNetCrossRefzbMATHGoogle Scholar
  27. 27.
    Kilgour, D. M., & Marshall, E. (2012). Approval balloting for fixed-size committees. In D. S. Felsenthal & M. Machover (Eds.), Electoral systems, studies in choice and welfare (pp. 305–326). Berlin: Springer.Google Scholar
  28. 28.
    Laslier, J. F., & Sanver, M. R. (Eds.). (2010). Handbook on approval voting. Berlin: Springer.zbMATHGoogle Scholar
  29. 29.
    Lenstra, H. W. (1983). Integer programming with a fixed number of variables. Mathematics of Operations Research, 8(4), 538–548.MathSciNetCrossRefzbMATHGoogle Scholar
  30. 30.
    Lin, A. P. (2011). The complexity of manipulating \(k\)-approval elections. ICAART, 2, 212–218. arXiv:1005.4159.Google Scholar
  31. 31.
    Meir, R., Procaccia, A. D., Rosenschein, J. S., & Zohar, A. (2008). Complexity of strategic behavior in multi-winner elections. Journal of Artificial Intelligence Research, 33, 149–178.MathSciNetCrossRefzbMATHGoogle Scholar
  32. 32.
    Miller, A. D. (2008). Group identification. Games and Economic Behavior, 63(1), 188–202.MathSciNetCrossRefzbMATHGoogle Scholar
  33. 33.
    Niedermeier, R. (2006). Invitation to fixed-parameter algorithms. Oxford: Oxford University Press Inc.CrossRefzbMATHGoogle Scholar
  34. 34.
    Samet, D., & Schmeidler, D. (2003). Between liberalism and democracy. Journal of Economic Theory, 110(2), 213–233.MathSciNetCrossRefzbMATHGoogle Scholar
  35. 35.
    West, D. B. (2000). Introduction to graph theory. Englewood Cliffs: Prentice-Hall.Google Scholar
  36. 36.
    Yang, Y., & Guo, J. (2014). Controlling elections with bounded single-peaked width. In AAMAS (pp. 629–636).Google Scholar
  37. 37.
    Yang, Y., & Guo, J. (2015). How hard is control in multi-peaked elections: A parameterized study. In AAMAS (pp. 1729–1730).Google Scholar
  38. 38.
    Yang, Y., & Guo, J. (2017). The control complexity of \(r\)-approval: From the single-peaked case to the general case. Journal of Computer and System Sciences, 89, 432–449.MathSciNetCrossRefzbMATHGoogle Scholar
  39. 39.
    Yang, Y., Wang, J. (2018). Multiwinner voting with restricted admissible sets: Complexity and strategyproofness. In IJCAI (to appear).Google Scholar

Copyright information

© The Author(s) 2018

Authors and Affiliations

  1. 1.Chair of Economic TheorySaarland UniversitySaarbrückenGermany

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