Advertisement

Robustness Radius for Chamberlin-Courant on Restricted Domains

  • Neeldhara MisraEmail author
  • Chinmay Sonar
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 11376)

Abstract

The notion of robustness in the context of committee elections was introduced by Bredereck et al. [SAGT 2018] [2] to capture the impact of small changes in the input preference orders, depending on the voting rules used. They show that for certain voting rules, such as Chamberlin-Courant, checking if an election instance is robust, even to the extent of a small constant, is computationally hard. More specifically, it is NP-hard to determine if one swap in any of the votes can change the set of winning committees with respect to the Chamberlin-Courant voting rule. Further, the problem is also \(\mathsf {W[1]}\)-hard when parameterized by the size of the committee, k. We complement this result by suggesting an algorithm that is in \(\mathsf {XP}\) with respect to k. We also show that on nearly-structured profiles, the problem of robustness remains NP-hard. We also address the case of approval ballots, where we show a hardness result analogous to the one established in [2] about rankings and again demonstrate an \(\mathsf {XP}\) algorithm.

Keywords

Robustness radius Chamberlin-Courant Single-peaked Single-crossing NP-hardness 

References

  1. 1.
    Brandt, F., Conitzer, V., Endriss, U., Lang, J., Procaccia, A.: Handbook of Computational Social Choice. Cambridge University Press, Cambridge (2016)CrossRefGoogle Scholar
  2. 2.
    Bredereck, R., Faliszewski, P., Kaczmarczyk, A., Niedermeier, R., Skowron, P., Talmon, N.: Robustness among multiwinner voting rules. In: Bilò, V., Flammini, M. (eds.) SAGT 2017. LNCS, vol. 10504, pp. 80–92. Springer, Cham (2017).  https://doi.org/10.1007/978-3-319-66700-3_7CrossRefGoogle Scholar
  3. 3.
    Chamberlin, J.R., Courant, P.N.: Representative deliberations and representative decisions: proportional representation and the Borda rule. Am. Polit. Sci. Rev. 77(03), 718–733 (1983)CrossRefGoogle Scholar
  4. 4.
    Cygan, M., et al.: Parameterized Algorithms. Springer, Cham (2015).  https://doi.org/10.1007/978-3-319-21275-3CrossRefzbMATHGoogle Scholar
  5. 5.
    Elkind, E., Faliszewski, P., Slinko, A.: Swap bribery. In: Mavronicolas, M., Papadopoulou, V.G. (eds.) SAGT 2009. LNCS, vol. 5814, pp. 299–310. Springer, Heidelberg (2009).  https://doi.org/10.1007/978-3-642-04645-2_27CrossRefGoogle Scholar
  6. 6.
    Endriss, U.: Trends in Computational Social Choice. lulu.com (2017)Google Scholar
  7. 7.
    Fleischner, H., Sabidussi, G., Sarvanov, V.I.: Maximum independent sets in 3- and 4-regular Hamiltonian graphs. Discrete Math. 310(20), 2742–2749 (2010). Graph Theory Dedicated to Carsten Thomassen on his 60th BirthdayMathSciNetCrossRefGoogle Scholar
  8. 8.
    Lackner, M., Skowron, P.: Consistent approval-based multi-winner rules. In: Proceedings of the 2018 ACM Conference on Economics and Computation, pp. 47–48. ACM (2018)Google Scholar
  9. 9.
    Misra, N., Sonar, C., Vaidyanathan, P.R.: On the complexity of Chamberlin-Courant on almost structured profiles. In: Rothe, J. (ed.) ADT 2017. LNCS (LNAI), vol. 10576, pp. 124–138. Springer, Cham (2017).  https://doi.org/10.1007/978-3-319-67504-6_9CrossRefGoogle Scholar
  10. 10.
    Shiryaev, D., Yu, L., Elkind, E.: On elections with robust winners. In: Proceedings of the International Conference on Autonomous Agents and Multi-Agent Systems, (AAMAS), pp. 415–422. IFAAMAS (2013)Google Scholar
  11. 11.
    Xia, L.: Computing the margin of victory for various voting rules. In: Proceedings of the ACM Conference on Electronic Commerce, (EC), pp. 982–999. ACM (2012)Google Scholar

Copyright information

© Springer Nature Switzerland AG 2019

Authors and Affiliations

  1. 1.Indian Institute of Technology, GandhinagarGandhinagarIndia

Personalised recommendations