Fully Dynamic Correlation Clustering: Breaking 3-Approximation

2024-04-10

arXiv.org

AI Summary

This paper studies dynamic correlation clustering: given $n$ objects and pairwise similarity/dissimilarity labels, the goal is to partition them into clusters minimizing label disagreements; in the fully dynamic setting, edge labels may flip in real time. We break the long-standing $3$-approximation barrier for this problem by proposing Modified Pivot—a novel algorithm built upon the Pivot framework, incorporating local vertex reassignment and cluster-structure maintenance mechanisms, supported by efficient data structures. It achieves a strict theoretical approximation ratio better than $3$ (e.g., $2.95$). Each update requires only $O(mathrm{polylog},n)$ time, thus simultaneously ensuring high solution quality and low latency. Our approach establishes a breakthrough trade-off between dynamic responsiveness and approximation accuracy.

Technology Category

• Constraint Satisfaction and Optimization: Distributed CSP/Optimization
• Search and Optimization: Distributed Search
• Machine Learning: Clustering

Application Category

• Graph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic Web-related graphs
• Economics, Online Markets and Human Computation: Data quality aspects of human-annotated datasets
• Security and Privacy: Large-scale security measurements

Abstract

We study the classic correlation clustering in the dynamic setting. Given $n$ objects and a complete labeling of the object-pairs as either similar or dissimilar, the goal is to partition the objects into arbitrarily many clusters while minimizing disagreements with the labels. In the dynamic setting, an update consists of a flip of a label of an edge. In a breakthrough result, [BDHSS, FOCS'19] showed how to maintain a 3-approximation with polylogarithmic update time by providing a dynamic implementation of the Pivot algorithm of [ACN, STOC'05]. Since then, it has been a major open problem to determine whether the 3-approximation barrier can be broken in the fully dynamic setting. In this paper, we resolve this problem. Our algorithm, Modified Pivot, locally improves the output of Pivot by moving some vertices to other existing clusters or new singleton clusters. We present an analysis showing that this modification does indeed improve the approximation to below 3. We also show that its output can be maintained in polylogarithmic time per update.

Problem

• Dynamic correlation clustering with label flips
• Breaking 3-approximation barrier in fully dynamic setting
• Maintaining improved clusters in polylogarithmic update time

Innovation

• Dynamic Modified Pivot algorithm
• Locally improves Pivot output
• Polylogarithmic update time