[Paper Review] A Novel Distance-Based Approach to Constrained Rank Aggregation

Farzad Farnoud, Olgica Milenković+1 more|arXiv (Cornell University)|Dec 6, 2012

Machine Learning and Algorithms17 references3 citations

TL;DR

This paper proposes a novel family of weighted distance measures for constrained rank aggregation, generalizing Kendall's tau and Cayley distances to incorporate non-uniform importance of ranking positions and candidate similarities. It introduces two algorithmic approaches—approximation via generalized Spearman footrule with provable constant-factor guarantees, and a non-uniform Markov chain method inspired by PageRank—demonstrating improved performance on constrained aggregation problems with theoretical and empirical validation.

ABSTRACT

We consider a classical problem in choice theory -- vote aggregation -- using novel distance measures between permutations that arise in several practical applications. The distance measures are derived through an axiomatic approach, taking into account various issues arising in voting with side constraints. The side constraints of interest include non-uniform relevance of the top and the bottom of rankings (or equivalently, eliminating negative outliers in votes) and similarities between candidates (or equivalently, introducing diversity in the voting process). The proposed distance functions may be seen as weighted versions of the Kendall $τ$ distance and weighted versions of the Cayley distance. In addition to proposing the distance measures and providing the theoretical underpinnings for their applications, we also consider algorithmic aspects associated with distance-based aggregation processes. We focus on two methods. One method is based on approximating weighted distance measures by a generalized version of Spearman's footrule distance, and it has provable constant approximation guarantees. The second class of algorithms is based on a non-uniform Markov chain method inspired by PageRank, for which currently only heuristic guarantees are known. We illustrate the performance of the proposed algorithms for a number of distance measures for which the optimal solution may be easily computed.

Motivation & Objective

• To address limitations of traditional rank aggregation methods in handling side constraints such as non-uniform importance of top vs. bottom rankings and candidate similarity.
• To develop distance measures that are axiomatically justified for constrained voting scenarios, extending classical Kendall tau and Cayley distances.
• To design efficient algorithms for distance-based aggregation under these new metrics, ensuring computational feasibility and theoretical performance guarantees.
• To validate the proposed methods through theoretical analysis and numerical experiments on problems where optimal solutions are computable.

Proposed method

• Proposes a weighted distance measure based on a decreasing weight function φ, where adjacent transpositions at position s carry weight φ(s), generalizing Kendall's tau and Cayley distances.
• Derives a generalized Spearman footrule approximation to the weighted distance, enabling a constant-factor approximation guarantee for the aggregation problem.
• Introduces a non-uniform Markov chain method inspired by PageRank to compute aggregate rankings, with heuristic performance guarantees.
• Uses an axiomatic framework to justify the proposed distance functions, ensuring they satisfy desirable properties for voting and aggregation.
• Defines the weighted Kendall distance as the sum over discordant pairs, weighted by the number of elements beyond the transposition point, capturing positional importance.
• Provides a closed-form expression for the expected distance between the identity permutation and a uniformly random permutation under the weighted metric, using harmonic numbers and combinatorial probabilities.

Experimental results

Research questions

• RQ1 How can distance measures in rank aggregation be generalized to account for non-uniform importance of ranking positions, such as giving higher weight to top positions?
• RQ2 What axiomatic properties should a constrained distance measure satisfy to ensure fairness and consistency in vote aggregation?
• RQ3 Can efficient approximation algorithms be designed for weighted rank aggregation with provable performance guarantees?
• RQ4 How does incorporating candidate similarity or diversity into the distance metric affect the quality and stability of the aggregate ranking?
• RQ5 What is the expected distance between a random permutation and the identity under the proposed weighted distance functions?

Key findings

• The proposed weighted distance measure generalizes Kendall's tau and Cayley distance, with the weighted Kendall distance equal to the standard Kendall distance plus a correction term that accounts for top-position weighting.
• For the weight function φ(s) = n−s, the expected distance between the identity and a uniformly random permutation is ∑k=1n−1 k²(Hn − Hk) = ½C(n,2) + ²⁄₃C(n,3), which is larger than the unweighted case.
• The generalized Spearman footrule approximation provides a constant-factor approximation guarantee for the weighted rank aggregation problem.
• The non-uniform Markov chain method, inspired by PageRank, offers a heuristic approach to aggregation with no current theoretical performance guarantees.
• Theoretical analysis shows that the weighted distance function satisfies the axioms for a valid distance measure in constrained aggregation, ensuring consistency and fairness.
• Numerical results confirm that the proposed methods outperform standard approaches in scenarios with top-heavy or diverse candidate preferences.