[Paper Review] Computation and Complexity of Preference Inference Based on Hierarchical Models

Nic Wilson, Anne-Marie George+1 more|arXiv (Cornell University)|Sep 17, 2024

Constraint Satisfaction and Optimization16 references7 citations

TL;DR

This paper studies inference of user preferences under hierarchical lexicographic models (HCLP). It proves coNP-completeness in general, and identifies polynomial-time cases when criterion importance forms a total order or fixed equivalence classes, with a polynomial algorithm for those cases.

ABSTRACT

Preference Inference involves inferring additional user preferences from elicited or observed preferences, based on assumptions regarding the form of the user's preference relation. In this paper we consider a situation in which alternatives have an associated vector of costs, each component corresponding to a different criterion, and are compared using a kind of lexicographic order, similar to the way alternatives are compared in a Hierarchical Constraint Logic Programming model. It is assumed that the user has some (unknown) importance ordering on criteria, and that to compare two alternatives, firstly, the combined cost of each alternative with respect to the most important criteria are compared; only if these combined costs are equal, are the next most important criteria considered. The preference inference problem then consists of determining whether a preference statement can be inferred from a set of input preferences. We show that this problem is coNP-complete, even if one restricts the cardinality of the equal-importance sets to have at most two elements, and one only considers non-strict preferences. However, it is polynomial if it is assumed that the user's ordering of criteria is a total ordering; it is also polynomial if the sets of equally important criteria are all equivalence classes of a given fixed equivalence relation. We give an efficient polynomial algorithm for these cases, which also throws light on the structure of the inference.

Motivation & Objective

• Motivate preference inference when user preferences are incomplete and elicited preferences must be extrapolated.
• Model alternatives with multi-criteria costs organized hierarchically (HCLP) and define a lexicographic-like comparison.
• Determine the computational complexity of inferring whether a preference statement follows from inputs under HCLP.
• Identify tractable special cases and provide efficient algorithms for them.

Proposed method

• Define HCLP structures with a set of alternatives, an associative-commutative monotonic cost operator ⊕, and a set of evaluations C.
• Construct HCLP models as ordered partitions of evaluations into levels, inducing a lexicographic-like preference order.
• Formalize satisfaction and inference: Γ ⊨φ iff every model satisfying Γ also satisfies φ, and relate to inconsistency via negation (Proposition 1).
• Prove coNP-completeness for HCLP-Deduction with C(t) when t>1 (Theorem 1) by reduction from 3-SAT (Proposition 2).
• Show polynomial-time solvability for the t=1 case, and for C(≡) (equivalence-class based) models (Theorem 1).
• Provide a polynomial algorithm for the total-order criterion case and discuss structural implications for inference.

Experimental results

Research questions

• RQ1 What is the computational complexity of inferring a preference statement under hierarchical-criterion models (HCLP)?
• RQ2 Are there tractable (polynomial-time) conditions for preference inference within HCLP, such as fixed equal-importance sizes or total orders of criteria?
• RQ3 How does restricting the structure of criterion importance (e.g., C(t) with small t or equivalence-class groupings) affect inference complexity?
• RQ4 What is the relationship between inference, satisfiability, and consistency in HCLP models?
• RQ5 Can we design efficient algorithms to decide consistency or deduction in practical special cases of HCLP?

Key findings

• The general preference inference problem under HCLP is coNP-complete, even with equal-importance sets limited to size at most two and only non-strict preferences.
• Inference becomes polynomial-time solvable when the user’s ordering of criteria is a total order.
• Inference is also polynomial when the equal-importance sets are equivalence classes of a fixed equivalence relation.
• An efficient polynomial algorithm is provided for the total-order case, shedding light on the inference structure.
• The paper provides a formal reduction from 3-SAT to establish hardness, and a constructive algorithmic scheme for tractable cases.