Skip to main content
SpringerLink
Log in

Aggregating multi-valued CP-nets: a CSP-based approach

  • Published:
Journal of Heuristics Aims and scope Submit manuscript

Abstract

We develop a framework for preference aggregation in multi-attribute, multi-valued domains, where agents’ preferences are represented by Conditional Preference Networks (CP-nets). Most existing work either does not consider computational requirements, or depends on the strong assumption that the agents can express their preferences by acyclic CP-nets that are compatible with a common order on the variables. In this paper, we focus on majoritarian aggregation of CP-nets. We propose a general approach that allows for aggregating preferences when the expressed CP-nets are not required to be acyclic. Moreover, there is no requirement for any common structure among the agents’ CP-nets. The proposed approach computes a set of locally winning alternatives through the reduction to a constraint satisfaction problem. We present results of experiments that demonstrate the efficiency and scalability of our approach. Through comprehensive experiments we also investigate the distributions of the numbers of locally winning alternatives with different CP-net structures, with varying domain sizes and varying numbers of variables and agents.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Fig. 1
Fig. 2
Fig. 3
Fig. 4
Fig. 5
Fig. 6
Fig. 7
Fig. 8
Fig. 9
Fig. 10

Similar content being viewed by others

Notes

  1. Another later work by Conitzer et al. (2011) further investigates the maximum and expected number of local Condorcet winners assuming a uniform distribution for the CP-nets.

  2. The Schwartz set is the union of all maximal mutually connected subsets. A maximal mutually connected subset is a subset of vertices such that there is a path between any two vertices in the set, but there is no path from a vertex outside the set to a vertex inside the set.

  3. With an odd number of complete CP-nets, the set of PLCWs is identical to the set of LCWs.

  4. Consider, for instance, the case where every preference relation in \(\mathbf {R}\) is empty. Then, \(NCW(\mathbf {R})=\emptyset \), while there exists \(\mathbf {R} \subseteq \mathbf {R}'\) where \(NCW(\mathbf {R}') \ne \emptyset \)

  5. cp-statement stands for ceteris paribus preference statement in this paper.

  6. When the CP tables are represented in a compact way, dominance testing in binary-valued acyclic CP-nets are PSPACE-complete (Goldsmith et al. 2008).

  7. When the domain variables are all binary (i.e., the case with binary CP-nets), the majority induced graph is also called a majority hypercube in Conitzer et al. (2011).

  8. Note that we only claim that the relation \(\succ _{\mathcal {H}}\) between neighbours is easy. Moreover, it does not necessarily mean that the majority relation \(\succ ^-_{maj}\) between neighbours is easy due to the possible incompleteness of CP-nets. As shown before, the majority induced graph does not show all the preference relations, even between neighbours: it might be the case that \(x\mathbf {w} \succ ^-_{maj} x'\mathbf {w}\) while \(x\mathbf {w} \nsucc _{\mathcal {H}} x'\mathbf {w}\) (for some \(x,x' \in D_X\) and \(\mathbf {w}\in D_{\mathbf {V}-\{X\}}\)).

  9. This is because when the CP-nets are incomplete, there exist profiles for which \(x\mathbf {w} \succ ^-_{maj}x'\mathbf {w}\) and \(x\mathbf {w} \nsucc _{\mathcal {H}} x'\mathbf {w}\) for some \(x,x'\in D_X\) and \(\mathbf {w} \in D_{\mathbf {V}-\{X\}}\).

  10. If the profile has just one agent, the majority preference is then equal to that agent’s preference. Therefore, the problem of winner determination is equal to the problem of individual outcome optimization. And as such, the corresponding CSP will be the same as the CSP in Brafman and Dimopoulos (2004) for single agent preference optimization.

  11. Here least literal means the literal with the smallest subscript in the clause.

  12. Interestingly, this result coincides with the analysis by Conitzer et al. (2011) even though we do not use a uniform distribution for generating the CP-nets in these experiments.

  13. To test dominance relations in acyclic CP-nets, the technique introduced in Li et al. (2011a) uses a numerical approximation of the given CP-net and considers the hamming distance between the currently considered outcome and the target outcome of the given query as a heuristic to guide the search process for an improving flipping sequence. As we have shown in Li et al. (2011a), this technique achieves significant efficiency gains.

References

  • Aziz, H., Harrenstein, P., Brill, M., Lang, J., Fischer, F., Seedig, G.H.: Possible and necessary winners of partial tournaments. In: Proceedings of the 11th International Conference on Autonomous Agents and Multiagent Systems, vol. 2, pp. 585–592. International Foundation for Autonomous Agents and Multiagent Systems (2012)

  • Bacchus, F., Grove, A.: Graphical models for preference and utility. In: Proceedings of the Eleventh Conference Annual Conference on Uncertainty in Artificial Intelligence (UAI-95), pp. 3–10. Morgan Kaufmann, San Francisco, CA (1995)

  • Boutilier, C., Brafman, R.I., Domshlak, C., Hoos, H.H., Poole, D.: CP-nets: a tool for representing and reasoning with conditional ceteris paribus preference statements. J. Artif. Int. Res. 21, 135–191 (2004)

    MATH  MathSciNet  Google Scholar 

  • Bouveret, S., Endriss, U., Lang, J.: Conditional importance networks: a graphical language for representing ordinal, monotonic preferences over sets of goods. In: IJCAI, pp. 67–72. (2009)

  • Brafman, R.I., Dimopoulos, Y.: Extended semantics and optimization algorithms for CP-networks. Comput. Intell. 20(2), 218–245 (2004)

    Article  MathSciNet  Google Scholar 

  • Brams, S.J., Marc Kilgour, D., Zwicker, W.S.: The paradox of multiple elections. Soc. Choice Welf. 15, 211–236 (1998)

    Article  MATH  Google Scholar 

  • Conitzer, V., Lang, J., Xia, L.: Hypercubewise preference aggregation in multi-issue domains. In: Proceedings of 22th IJCAI, pp. 158–163. International Joint Conference on Artificial Intelligence, Barcelona, Spain, July (2011)

  • Dimopoulos, Y., Michael, L., Athienitou, F.: Ceteris paribus preference elicitation with predictive guarantees. Proceedings of the 21st International Jont conference on Artifical intelligence. IJCAI’09, pp. 1890–1895. Morgan Kaufmann Publishers Inc, San Francisco, CA, USA (2009)

  • Domshlak, C., Brafman, R.I.: CP-nets-reasoning and consistency testing. In: KR, pp. 121–132. (2002)

  • East, D., Truszczyński, M.: Propositional satisfiability in answer-set programming. In: KI 2001: Advances in Artificial Intelligence, pp. 138–153. Springer (2001)

  • Farquharson, R.: Theory of Voting. Yale University Press, New Haven (1969)

    Google Scholar 

  • Goldsmith, J., Lang, J., Truszczynski, M., Wilson, N.: The computational complexity of dominance and consistency in CP-nets. J. Artif. Int. Res. 33(1), 403–432 (2008)

    MATH  MathSciNet  Google Scholar 

  • Konczak, K., Lang, J.: Voting procedures with incomplete preferences. In: Brafman, R., Junker, U. (eds.), IJCAI-05 Workshop on Advances in Preference Handling, pp. 124–129. (2005)

  • Koriche, F., Zanuttini, B.: Learning conditional preference networks. Artif. Intell. 174(11), 685–703 (2010)

    Article  MATH  MathSciNet  Google Scholar 

  • Lang, J., Xia, L.: Sequential composition of voting rules in multi-issue domains. Math. Soc. Sci. 57(3), 304–324 (2009)

    Article  MATH  MathSciNet  Google Scholar 

  • Li, M., Vo, Q.B., Kowalczyk, R.: An efficient majority-rule-based approach for collective decision making with CP-nets. In: Proceedings of the Twelfth International Conference on the Principles of Knowledge Representation and Reasoning, pp. 578–580. Toronto, Ontario, Canada (2010)

  • Li, M., Vo, Q.B., Kowalczyk, R.: Efficient heuristic approach to dominance testing in CP-nets. Proceedings of the 10th International Conference on Autonomous Agents and Multiagent Systems. vol. 1, pp. 353–360. International Foundation for Autonomous Agents and Multiagent Systems, Taipei, Taiwan, Richland, SC (2011a)

  • Li, M., Vo, Q.Bao, Kowalczyk, R.: Majority-rule-based preference aggregation on multi-attribute domains with CP-nets. In: Proceedings of the 10th International Conference on Autonomous Agents and Multiagent Systems, vol. 1, pp. 659–666. International Foundation for Autonomous Agents and Multiagent Systems, Taipei, Taiwan, Richland, SC (2011b)

  • Liu, L., Truszczyński, M.: Local-search techniques for propositional logic extended with cardinality constraints. In: Principles and Practice of Constraint Programming-CP 2003, pp. 495–509. Springer (2003)

  • Miguel, I., Shen, Q.: Solution techniques for constraint satisfaction problems: advanced approaches. Artif. Intell. Rev. 15(4), 269–293 (2001)

    Article  MATH  Google Scholar 

  • Pozza, G.D., Pini, M.S., Rossi, F., V., Kristen B.: Multi-agent soft constraint aggregation via sequential voting. In: IJCAI, pp. 172–177. (2011)

  • Rossi, F., Venable, K.B., Walsh, T.: mCP nets: representing and reasoning with preferences of multiple agents. In: McGuinness, D.L., Ferguson, G. (eds.) AAAI, pp. 729–734. AAAI Press/The MIT Press, (2004)

  • Tovey, C.A.: A simplified NP-complete satisfiability problem. Discret. Appl. Math. 8(1), 85–89 (1984)

    Article  MATH  MathSciNet  Google Scholar 

  • Xia, L., Conitzer, V.: Determining possible and necessary winners given partial orders. J. Artif. Intell. Res. (JAIR) 41, 25–67 (2011)

    MATH  MathSciNet  Google Scholar 

  • Xia, L., Lang, J., Ying, M.: Strongly decomposable voting rules on multiattribute domains. In: AAAI’07: Proceedings of the 22nd National Conference on Artificial intelligence, pp. 776–781. AAAI Press (2007)

  • Xia, L., Conitzer, V., Lang, J.: Voting on multiattribute domains with cyclic preferential dependencies. In: AAAI’08: Proceedings of the 23rd National Conference on Artificial intelligence, pp. 202–207. AAAI Press (2008)

Download references

Acknowledgments

We thank Jérôme Lang, Lirong Xia, Toby Walsh and the anonymous reviewers for KR2010, AAMAS2011, Journal of AI for fruitful discussions and comments. This work was supported by the ARC Discovery Grants DP110103671. We would also like to thank the three anonymous reviewers for the Journal of Heuristics who provide very thorough comments and suggestions.

Author information

Authors and Affiliations

  1. Swinburne University of Technology, Hawthorn, Australia

    Minyi Li, Quoc Bao Vo & Ryszard Kowalczyk

Authors
  1. Minyi Li
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Quoc Bao Vo
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Ryszard Kowalczyk
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Minyi Li.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Li, M., Vo, Q.B. & Kowalczyk, R. Aggregating multi-valued CP-nets: a CSP-based approach. J Heuristics 21, 107–140 (2015). https://doi.org/10.1007/s10732-014-9276-8

Download citation

  • Received:

  • Revised:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10732-014-9276-8

Keywords

Search

Navigation