Online Correlation Clustering

TL;DR

This paper investigates online correlation clustering, analyzing the performance of greedy algorithms for minimizing disagreements and maximizing agreements, establishing optimal competitive ratios and proposing improved randomized algorithms.

Contribution

It provides tight bounds for greedy algorithms in online correlation clustering and introduces a randomized approach that surpasses the 0.5 competitive ratio barrier.

Findings

Greedy algorithm is O(n)-competitive for minimizing disagreements.

Greedy algorithm is 0.5-competitive for maximizing agreements.

A randomized algorithm achieves slightly better than 0.5 competitiveness.

Abstract

We study the online clustering problem where data items arrive in an online fashion. The algorithm maintains a clustering of data items into similarity classes. Upon arrival of v, the relation between v and previously arrived items is revealed, so that for each u we are told whether v is similar to u. The algorithm can create a new cluster for v and merge existing clusters. When the objective is to minimize disagreements between the clustering and the input, we prove that a natural greedy algorithm is O(n)-competitive, and this is optimal. When the objective is to maximize agreements between the clustering and the input, we prove that the greedy algorithm is .5-competitive; that no online algorithm can be better than .834-competitive; we prove that it is possible to get better than 1/2, by exhibiting a randomized algorithm with competitive ratio .5+c for a small positive fixed constant c.