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Towards Fast Algorithms for the Preference Consistency Problem Based on Hierarchical Models

Anne-Marie George, Nic Wilson, Barry O'Sullivan

TL;DR

Three approaches to solve the Preference Consistency Problem for preference statements based on hierarchical models are developed, the first involves a Mixed Integer Linear Programming (MILP) formulation, and the other two are recursive algorithms that are based on properties of the problem by which the search space can be pruned.

Abstract

In this paper, we construct and compare algorithmic approaches to solve the Preference Consistency Problem for preference statements based on hierarchical models. Instances of this problem contain a set of preference statements that are direct comparisons (strict and non-strict) between some alternatives, and a set of evaluation functions by which all alternatives can be rated. An instance is consistent based on hierarchical preference models, if there exists an hierarchical model on the evaluation functions that induces an order relation on the alternatives by which all relations given by the preference statements are satisfied. Deciding if an instance is consistent is known to be NP-complete for hierarchical models. We develop three approaches to solve this decision problem. The first involves a Mixed Integer Linear Programming (MILP) formulation, the other two are recursive algorithms that are based on properties of the problem by which the search space can be pruned. Our experiments on synthetic data show that the recursive algorithms are faster than solving the MILP formulation and that the ratio between the running times increases extremely quickly.

Towards Fast Algorithms for the Preference Consistency Problem Based on Hierarchical Models

TL;DR

Three approaches to solve the Preference Consistency Problem for preference statements based on hierarchical models are developed, the first involves a Mixed Integer Linear Programming (MILP) formulation, and the other two are recursive algorithms that are based on properties of the problem by which the search space can be pruned.

Abstract

In this paper, we construct and compare algorithmic approaches to solve the Preference Consistency Problem for preference statements based on hierarchical models. Instances of this problem contain a set of preference statements that are direct comparisons (strict and non-strict) between some alternatives, and a set of evaluation functions by which all alternatives can be rated. An instance is consistent based on hierarchical preference models, if there exists an hierarchical model on the evaluation functions that induces an order relation on the alternatives by which all relations given by the preference statements are satisfied. Deciding if an instance is consistent is known to be NP-complete for hierarchical models. We develop three approaches to solve this decision problem. The first involves a Mixed Integer Linear Programming (MILP) formulation, the other two are recursive algorithms that are based on properties of the problem by which the search space can be pruned. Our experiments on synthetic data show that the recursive algorithms are faster than solving the MILP formulation and that the ratio between the running times increases extremely quickly.

Paper Structure

This paper contains 8 sections, 6 theorems, 13 equations, 3 figures, 3 tables.

Table of Contents

  1. Introduction
  2. The Consistency Problem Based on Hierarchical Models
  3. Hierarchical Models
  4. Preference Consistency
  5. MILP Formulation
  6. Recursive Algorithms
  7. Experimental Runtime Comparisons
  8. Conclusion

Key Result

Proposition 2.5

$H \vDash^{\oplus} \varphi$ for all $H \in {\cal C}(t)$ with $H \vDash^{\oplus} \Gamma$ if and only if $\Gamma\cup \{\neg\varphi\}$ is ${\cal C}(t)$-inconsistent, where $\neg\varphi = \beta_\varphi <~\alpha_\varphi$ for $\varphi \in {\cal L}^{{\cal A}}_{\leq}$, or $\neg\varphi = \beta_\varphi \leq \

Figures (3)

  • Figure 1: Mean times in seconds of PC-check(${\cal C}, \Gamma, \oplus, t$) (left) and PC-check(${\cal C}, \Gamma, \oplus, t, {\cal S}, s$) (right).
  • Figure 2: Percentages of instance classes for the tested random instances with $n=30$ (top) and $n=35$ (bottom).
  • Figure :

Theorems & Definitions (16)

  • Definition 2.1: HCLP structures
  • Definition 2.2: HCLP models
  • Example
  • Definition 2.3: Satisfaction of Preference Statements
  • Definition 2.4: Consistency
  • Example : continued
  • Proposition 2.5
  • Lemma 4.1
  • proof
  • Proposition 4.2
  • ...and 6 more