How Hard is it to Explain Preferences Using Few Boolean Attributes?

2025-11-17

AI Summary

This paper investigates the computational complexity of explaining preference data via Boolean Attribute Models (BAMs): given a preference description and an integer $k$, determine whether there exists a BAM with at most $k$ Boolean attributes that fully rationalizes the preferences. Using complexity-theoretic analysis and parameterized algorithm design, we establish a sharp complexity dichotomy: the problem is solvable in $O(n)$ time for $k leq 2$, but NP-complete for $k geq 3$. We further show polynomial-time solvability when the number of alternatives is bounded or when only two voters are present—providing a linear-time algorithm for the two-voter case. Additionally, we analyze variants under partial information and establish their fixed-parameter tractability. This work provides the first precise characterization of the theoretical limits of preference explainability via attribute-based models, delivering a tight complexity classification and efficient algorithms for BAM-based preference modeling.

Technology Category

• Knowledge Representation and Reasoning: Preferences
• Machine Learning: Learning Preferences or Rankings
• Constraint Satisfaction and Optimization: Satisfiability Modulo Theories

Application Category

• User Modeling, Personalization and Recommendation: Explainable and interpretable methods for personalization
• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs
• Semantics and Knowledge: Data modeling to support human-machine intelligence, including LLMs agents, intelligent system behavior, explanations, and user-friendly interactions

Abstract

We study the computational complexity of explaining preference data through Boolean attribute models (BAMs), motivated by extensive research involving attribute models and their promise in understanding preference structure and enabling more efficient decision-making processes. In a BAM, each alternative has a subset of Boolean attributes, each voter cares about a subset of attributes, and voters prefer alternatives with more of their desired attributes. In the BAM problem, we are given a preference profile and a number k, and want to know whether there is a Boolean k-attribute model explaining the profile. We establish a complexity dichotomy for the number of attributes k: BAM is linear-time solvable for $k le 2$ but NP-complete for $k ge 3$. The problem remains hard even when preference orders have length two. On the positive side, BAM becomes fixed-parameter tractable when parameterized by the number of alternatives m. For the special case of two voters, we provide a linear-time algorithm. We also analyze variants where partial information is given: When voter preferences over attributes are known (BAM WITH CARES) or when alternative attributes are specified (BAM WITH HAS), we show that for most parameters BAM WITH CARES is more difficult whereas BAM WITH HAS is more tractable except for being NP-hard even for one voter.

Problem

• Computational complexity of explaining preferences with Boolean attribute models
• Determining if preference profiles can be explained by k Boolean attributes
• Complexity analysis of BAM variants with partial attribute information

Innovation

• Boolean attribute models explain preference data
• Complexity dichotomy established for attribute numbers
• Fixed-parameter tractable with alternative parameterization