Optimizing Weak Orders via Integer Linear Programming

TL;DR

The paper develops a unified Integer Linear Programming framework to exactly solve rank-aggregation problems where the consensus is a weak (bucket) order, with OBOP as a central case. It provides complete ILP formulations for OBOP, its variants with a fixed number of buckets, Tail-Collapsed Upper-$k$, and fairness constraints, along with the utopian matrix bound. Computational experiments on PrefLib and related benchmarks demonstrate that the exact formulations achieve optimal solutions for moderate-sized instances and offer precise benchmarks against state-of-the-art heuristics. The work contributes rigorous formulations, practical variants, and fairness-aware mechanisms, establishing a foundation for exact, scalable, and equitable rank aggregation under weak orders.

Abstract

Rank aggregation problems aim to combine multiple individual orderings of a common set of items into a consensus ranking that best reflects the collective preferences. This paper introduces a general Integer Linear Programming (ILP) framework to model and solve, in an exact way, problems whose solutions are weak orders (a.k.a.\ bucket orders). Within this framework, we consider additional relevant constraints to produce the consensus bucket order, considering configurations with a fixed number of buckets, predefined bucket sizes, top-$k$ type problems, and fairness constraints. All these formulations are developed in a general setting, allowing their application to different rank aggregation contexts. One of these problems is the Optimal Bucket Order Problem (OBOP), for which we propose for the first time an exact formulation and test the variants proposed. The computational study includes, on the one hand, a comparison between the exact results obtained by our models and the heuristic methods proposed by Aledo et al.\ (2018), and on the other hand, an additional evaluation of their performance on a representative set of instances from the PrefLib library. The results confirm the validity and efficiency of the proposed approach, providing a solid foundation for future research on rank aggregation problems with weak orders as consensus rankings.

Figures (7)

• Figure 1: Pairwise order matrix used in Example \ref{['ex:ejemplo_1']}
• Figure 2: Pairwise order matrix used in Example \ref{['ex:contraejemplo_top-k']}
• Figure 3: Plot of the $p$-OBOP results across instances BOX-42-01 to BOX-42-05
• Figure 4: Plot of the TCU-$k$ OBOP results across instances BOX-42-01 to BOX-42-05
• Figure 5: Comparison of fairness and non-fairness results for instance BOX-42-10

Theorems & Definitions (26)

• Proposition 1
• proof
• Corollary 1
• proof
• Remark 1
• Example 1
• Proposition 2
• proof
• Proposition 3
• proof