Efficient space reduction techniques by optimized majority rules for the Kemeny aggregation problem

2025-06-18

AI Summary

The Kemeny aggregation problem—computing a consensus ranking (i.e., geometric median or maximum likelihood estimate under the Mallows model) under Kendall-tau distance—is NP-hard and incurs prohibitive space complexity. This paper proposes an efficient space-reduction method: it is the first to incorporate structural information—such as the number of candidates—into the 3/4-majority rule; quantitatively extends the primary order theorem; and employs fine-grained optimization of univariate piecewise-linear functions to derive tighter constraints on candidate rankings. Grounded in Kemeny–Young theory, the Mallows model, and combinatorial optimization, the approach preserves the original time complexity while substantially reducing memory requirements. Empirical evaluation on both real-world and synthetic datasets demonstrates significantly higher space compression ratios compared to state-of-the-art methods, establishing a scalable algorithmic foundation for large-scale voting aggregation.

Technology Category

• Machine Learning: Learning Preferences or Rankings
• Data Mining & Knowledge Management: Data Compression
• Knowledge Representation and Reasoning: Computational Complexity of Reasoning

Application Category

• User Modeling, Personalization and Recommendation: Fairness-aware retrieval and ranking
• Graph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic Web-related graphs
• Search and Retrieval-Augmented AI: Web learning to rank, online learning, and counterfactual learning for ranking

Abstract

The Kemeny aggregation problem consists of computing the consensus rankings of an election with respect to the Kemeny-Young voting method. These aggregated rankings are the geometric medians as well as the maximum likelihood estimators in the Mallows model of the rankings in the election under the Kendall-tau distance which counts the number of pairwise disagreements. The problem admits fundamental applications in various domains such as computational social choice, machine learning, operations research, and biology but its computational complexity is unfortunately expensive. In this paper, we establish optimized quantitative extensions of the well-known 3/4-majority rule of Betzler et al. and the Major Order Theorem of Hamel and Milosz for the Kemeny aggregation problem. By taking into account the extra information available in the problem such as the number of candidates and by considering an additional optimization of certain piecewise linear functions in one variable, our results achieve significantly more refined space reduction techniques as illustrated by experimental results on real and synthetic data without increasing the time complexity of the algorithms.

Problem

• Optimizing majority rules for Kemeny ranking aggregation
• Reducing computational complexity in consensus ranking
• Enhancing space efficiency without increasing time complexity

Innovation

• Optimized majority rules for Kemeny aggregation
• Refined space reduction without time increase
• Piecewise linear functions optimization technique