A Scalable and Unified Framework to Weighted Rank Aggregation

TL;DR

This paper introduces a scalable, unified framework for weighted rank aggregation across multiple distance metrics, providing new approximation algorithms that are efficient in massively parallel computation environments.

Contribution

It develops a general structural principle for rank aggregation, and presents novel MPC algorithms with improved approximation guarantees for various metrics.

Findings

A unified framework reduces rank aggregation to local median problems.

New MPC algorithms achieve constant-round approximations with sublinear local memory.

Improved approximation ratio of 1.968 for Ulam distance extends to weighted cases.

Abstract

The rank aggregation problem seeks to combine multiple rank orderings of the same set of candidates into a single consensus ordering. Such problems arise in diverse domains, including web search, employment, college admissions, and voting. In this work we focus on the 1-median objective: given a set of m rankings over [n], the goal is to compute a ranking that minimizes the sum of its distances to all input rankings. We study rank aggregation under several classical distance metrics: Ulam distance, Spearman's footrule, Hamming distance, and Kendall-tau, as well as their weighted variants. Our contributions begin with a novel unified framework that identifies a key structural property: it suffices to focus on a small subset of rankings, where the corresponding local one-median provides a good approximation to the global median. This principle extends across these distance measures, yielding a general algorithmic framework for weighted rank aggregation. Building on this, we present a new approximation algorithm for rank aggregation under the Ulam distance that scales in the Massively Parallel Computation (MPC) model. Our algorithm computes a $(2-\alpha)$-approximation, for a constant $\alpha>0$, to the 1-median in a constant number of rounds, using local memory sublinear in n and total memory near-linear in n. We further design new MPC approximation algorithms for Spearman's footrule and for the element-weighted variants of Hamming and Kendall-tau distances. For each metric, we obtain a $(2-\zeta)$-approximation, for a constant $\zeta>0$, to the 1-median in a constant number of rounds, using local memory sublinear in n and total memory linear or near-linear in n. Moreover, for the Ulam distance, we simplify and strengthen the analysis of Chakraborty et al., obtaining an improved 1.968-approximation that further extends to the weighted setting.