Abstract
Preference trees, or P-trees for short, offer an intuitive and often concise way of representing preferences over combinatorial domains. In this paper, we propose an alternative definition of P-trees, and formally introduce their compact representation that exploits occurrences of identical subtrees. We show that P-trees generalize lexicographic preference trees and are strictly more expressive. We relate P-trees to answer-set optimization programs and possibilistic logic theories. Finally, we study reasoning with P-trees and establish computational complexity results for the key reasoning tasks of comparing outcomes with respect to orders defined by P-trees, and of finding optimal outcomes.
This is a preview of subscription content, log in via an institution to check access.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Similar content being viewed by others
Notes
- 1.
We overload the symbols \(\succeq _T\) and \(\succ _T\) by using them both for the order on the leaves of T and the corresponding preorder on the outcomes from \( CD (\mathcal {I})\).
- 2.
This definition is a slight adaptation of the original one.
- 3.
Given a Boolean formula \(\varPhi \) over \(\{x_1,\ldots ,x_n\}\), the maximum satisfying assignment (MSA) problem is to decide whether \(x_n=1\) in the lexicographically maximum satisfying assignment for \(\varPhi \). (If \(\varPhi \) is unsatisfiable, the answer is \(\textit{no}\).).
References
Booth, R., Chevaleyre, Y., Lang, J., Mengin, J., Sombattheera, C.: Learning conditionally lexicographic preference relations. In: ECAI, pp. 269–274 (2010)
Boutilier, C., Brafman, R., Domshlak, C., Hoos, H., Poole, D.: CP-nets: a tool for representing and reasoning with conditional ceteris paribus preference statements. J. Artif. Intell. Res. 21, 135–191 (2004)
Brewka, G., Niemelä, I., Truszczynski, M.: Answer set optimization. In: IJCAI, pp. 867–872 (2003)
Dubois, D., Lang, J., Prade, H.: A brief overview of possibilistic logic. In: ECSQARU, pp. 53–57 (1991)
Fraser, N.M.: Applications of preference trees. In: Proceedings of IEEE Systems Man and Cybernetics Conference, pp. 132–136. IEEE (1993)
Fraser, N.M.: Ordinal preference representations. Theory Decis. 36(1), 45–67 (1994)
Garey, M.R., Johnson, D.S.: Computers and Intractability: A Guide to the Theory of NP-Completeness. W. H. Freeman & Co., New York (1979)
Kaci, S.: Working with Preferences: Less Is More. Cognitive Technologies. Springer, Berlin (2011)
Krentel, M.W.: The complexity of optimization problems. J. Comput. Syst. Sci. 36(3), 490–509 (1988)
Lang, J., Mengin, J., Xia, L.: Aggregating conditionally lexicographic preferences on multi-issue domains. In: CP, pp. 973–987 (2012)
Liu, X., Truszczynski, M.: Aggregating Conditionally lexicographic preferences using answer set programming solvers. In: Perny, P., Pirlot, M., Tsoukià s, A. (eds.) ADT 2013. LNCS, vol. 8176, pp. 244–258. Springer, Heidelberg (2013)
Author information
Authors and Affiliations
Corresponding author
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2015 Springer International Publishing Switzerland
About this paper
Cite this paper
Liu, X., Truszczynski, M. (2015). Reasoning with Preference Trees over Combinatorial Domains. In: Walsh, T. (eds) Algorithmic Decision Theory. ADT 2015. Lecture Notes in Computer Science(), vol 9346. Springer, Cham. https://doi.org/10.1007/978-3-319-23114-3_2
Download citation
DOI: https://doi.org/10.1007/978-3-319-23114-3_2
Published:
Publisher Name: Springer, Cham
Print ISBN: 978-3-319-23113-6
Online ISBN: 978-3-319-23114-3
eBook Packages: Computer ScienceComputer Science (R0)