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

TL;DR

This work formalizes the problem of explaining human preferences using a parsimonious set of Boolean attributes (BAM) and analyzes its computational complexity. It establishes a sharp dichotomy: the BAM decision problem is solvable in linear time for at most $k\le 2$ attributes but becomes NP-hard for $k\ge 3$, even when preference lists are short, and shows that the problem is fixed-parameter tractable when parameterized by the number of alternatives $m$, with efficient results for the two-voter case. The authors also study variants where partial information is provided (BAM with Cares and BAM with Has), showing that the Cares variant is typically harder while the Has variant is more tractable in many regimes, though NP-hard even for a single voter. Collectively, these results map a nuanced landscape of tractability and hardness that informs algorithmic approaches for learning and explaining attribute-based preferences, and highlight promising directions for structural parameterization and ILP-based methods. The findings have practical implications for scalable preference elicitation, decision making, and AI explainability in domains where binary feature models are natural.

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.

Theorems & Definitions (61)

• Remark 1
• Lemma 1
• Lemma 1
• proof
• Example 1
• Lemma 2
• Lemma 2
• proof
• Example 2
• Example 3