Rank Aggregation: New Bounds for MCx

TL;DR

This paper investigates the theoretical performance bounds of Markov chain-based rank aggregation algorithms, establishing new lower bounds and relating them to classical scoring methods.

Contribution

It provides the first known lower bounds for the approximation guarantees of MC1, MC2, MC3, MC4 algorithms and connects MC4 to Copeland score.

Findings

Established supra-constant lower bounds for MC algorithms

Raised the lower bound for sorting by Copeland score to 2

Showed MC4 as a generalization of Copeland score

Abstract

The rank aggregation problem has received significant recent attention within the computer science community. Its applications today range far beyond the original aim of building metasearch engines to problems in machine learning, recommendation systems and more. Several algorithms have been proposed for these problems, and in many cases approximation guarantees have been proven for them. However, it is also known that some Markov chain based algorithms (MC1, MC2, MC3, MC4) perform extremely well in practice, yet had no known performance guarantees. We prove supra-constant lower bounds on approximation guarantees for all of them. We also raise the lower bound for sorting by Copeland score from 3/2 to 2 and prove an upper bound of 11, before showing that in particular ways, MC4 can nevertheless be seen as a generalization of Copeland score.