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A consensus set for the aggregation of partial rankings: the case of the Optimal Set of Bucket Orders Problem

Juan A. Aledo, José A. Gámez, Alejandro Rosete

TL;DR

The paper tackles rank aggregation by shifting from a single consensus to a weighted set of bucket orders (OSBOP), enabling the capture of diversity in input preferences. It generalizes OBOP via a formal OSBOP framework, introduces utopian matrices for guidance, and deploys a two-level stochastic local search to find both bucket orders and their weights. Empirical results on PrefLib data show that OSBOP substantially improves fit to input preferences over OBOP, especially when weight flexibility is allowed, while maintaining interpretability. This approach offers a practical path to fairer, multi-view ranking that reflects minority viewpoints and distinct communities within the data.

Abstract

In rank aggregation problems (RAP), the solution is usually a consensus ranking that generalizes a set of input orderings. There are different variants that differ not only in terms of the type of rankings that are used as input and output, but also in terms of the objective function employed to evaluate the quality of the desired output ranking. In contrast, in some machine learning tasks (e.g. subgroup discovery) or multimodal optimization tasks, attention is devoted to obtaining several models/results to account for the diversity in the input data or across the search landscape. Thus, in this paper we propose to provide, as the solution to an RAP, a set of rankings to better explain the preferences expressed in the input orderings. We exemplify our proposal through the Optimal Bucket Order Problem (OBOP), an RAP which consists in finding a single consensus ranking (with ties) that generalizes a set of input rankings codified as a precedence matrix. To address this, we introduce the Optimal Set of Bucket Orders Problem (OSBOP), a generalization of the OBOP that aims to produce not a single ranking as output but a set of consensus rankings. Experimental results are presented to illustrate this proposal, showing how, by providing a set of consensus rankings, the fitness of the solution significantly improves with respect to the one of the original OBOP, without losing comprehensibility.

A consensus set for the aggregation of partial rankings: the case of the Optimal Set of Bucket Orders Problem

TL;DR

The paper tackles rank aggregation by shifting from a single consensus to a weighted set of bucket orders (OSBOP), enabling the capture of diversity in input preferences. It generalizes OBOP via a formal OSBOP framework, introduces utopian matrices for guidance, and deploys a two-level stochastic local search to find both bucket orders and their weights. Empirical results on PrefLib data show that OSBOP substantially improves fit to input preferences over OBOP, especially when weight flexibility is allowed, while maintaining interpretability. This approach offers a practical path to fairer, multi-view ranking that reflects minority viewpoints and distinct communities within the data.

Abstract

In rank aggregation problems (RAP), the solution is usually a consensus ranking that generalizes a set of input orderings. There are different variants that differ not only in terms of the type of rankings that are used as input and output, but also in terms of the objective function employed to evaluate the quality of the desired output ranking. In contrast, in some machine learning tasks (e.g. subgroup discovery) or multimodal optimization tasks, attention is devoted to obtaining several models/results to account for the diversity in the input data or across the search landscape. Thus, in this paper we propose to provide, as the solution to an RAP, a set of rankings to better explain the preferences expressed in the input orderings. We exemplify our proposal through the Optimal Bucket Order Problem (OBOP), an RAP which consists in finding a single consensus ranking (with ties) that generalizes a set of input rankings codified as a precedence matrix. To address this, we introduce the Optimal Set of Bucket Orders Problem (OSBOP), a generalization of the OBOP that aims to produce not a single ranking as output but a set of consensus rankings. Experimental results are presented to illustrate this proposal, showing how, by providing a set of consensus rankings, the fitness of the solution significantly improves with respect to the one of the original OBOP, without losing comprehensibility.

Paper Structure

This paper contains 14 sections, 24 equations, 2 figures, 3 tables.

Table of Contents

  1. Introduction
  2. Set up
  3. Bucket orders
  4. The Optimal Bucket Order Problem (OBOP)
  5. The Optimal Set of Bucket Orders Problem (OSBOP)
  6. Size of the solution space
  7. The utopian matrix for OSBOP
  8. Illustrative examples of OSBOP vs OBOP
  9. A simple metaheuristic approach for OSBOP
  10. Experimental results
  11. Results
  12. Best solutions found for the OSBOP
  13. Discussion
  14. Conclusions

Figures (2)

  • Figure 1: Size of the solution spaces for different variants of OSBOP.
  • Figure 2: A local search approach for OSBOP

Theorems & Definitions (4)

  • Definition 3.1: OSBOP
  • Example 3.1
  • Remark 3.1
  • Example 3.2