Online Correlation Clustering: Simultaneously Optimizing All $ell_p$-norms

2025-10-16

AI Summary

This work addresses the challenge of simultaneously optimizing all $ell_p$-norms ($p geq 1$) in online correlation clustering—a problem for which unified approximation algorithms exist offline, but only norm-specific online algorithms are known. We propose the first single algorithm for the online-with-a-sample (AOS) model that achieves unified competitive ratios across all $ell_p$-norms: $O(log^4 n)$ with high probability for $p < infty$, $O(log n)$ for $p = infty$, and $O(1)$ in expectation for $ell_1$. These guarantees are shown to be near-optimal via matching lower bounds. Our results uncover an inherent trade-off among multiple objectives in the random-order model and advance the theory of online clustering beyond worst-case analysis.

Technology Category

• Machine Learning: Online Learning & Bandits
• Search and Optimization: Algorithm Configuration
• Constraint Satisfaction and Optimization: Distributed CSP/Optimization

Application Category

• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs
• Economics, Online Markets and Human Computation: Data quality aspects of human-annotated datasets
• Responsible Web: Human-perceived consequences of algorithmic deployment on the web

Abstract

The $ell_p$-norm objectives for correlation clustering present a fundamental trade-off between minimizing total disagreements (the $ell_1$-norm) and ensuring fairness to individual nodes (the $ell_infty$-norm). Surprisingly, in the offline setting it is possible to simultaneously approximate all $ell_p$-norms with a single clustering. Can this powerful guarantee be achieved in an online setting? This paper provides the first affirmative answer. We present a single algorithm for the online-with-a-sample (AOS) model that, given a small constant fraction of the input as a sample, produces one clustering that is simultaneously $O(log^4 n)$-competitive for all $ell_p$-norms with high probability, $O(log n)$-competitive for the $ell_infty$-norm with high probability, and $O(1)$-competitive for the $ell_1$-norm in expectation. This work successfully translates the offline "all-norms" guarantee to the online world. Our setting is motivated by a new hardness result that demonstrates a fundamental separation between these objectives in the standard random-order (RO) online model. Namely, while the $ell_1$-norm is trivially $O(1)$-approximable in the RO model, we prove that any algorithm in the RO model for the fairness-promoting $ell_infty$-norm must have a competitive ratio of at least $Ω(n^{1/3})$. This highlights the necessity of a different beyond-worst-case model. We complement our algorithm with lower bounds, showing our competitive ratios for the $ell_1$- and $ell_infty$- norms are nearly tight in the AOS model.

Problem

• Simultaneously approximating all ℓp-norms in online correlation clustering
• Bridging offline all-norms guarantee to online computational settings
• Overcoming fundamental separation between ℓ1 and ℓ∞ objectives online

Innovation

• Online algorithm simultaneously optimizes all correlation clustering norms
• Uses small input sample to achieve competitive clustering guarantees
• Translates offline all-norms approximation to online setting