Online Correlation Clustering with Metric Weights

2026-08-06

AI Summary

This work addresses online correlation clustering under adversarial arrival orders, a setting where achieving sublinear competitive ratios is typically impossible due to an Ω(n) lower bound. Focusing on the variant with metric weights—where edge weights satisfy the probabilistic constraint \(w^+ + w^- = 1\) and a triangle inequality for negative weights—the paper presents the first fully online, deterministic algorithm that attains a constant competitive ratio against adversarial inputs. Specifically, the total weighted disagreement cost incurred by the algorithm is at most an \(O(1)\) factor greater than that of the optimal offline solution. This result demonstrates that metric consistency is the key structural property enabling tractability in this context and provides, for the first time, a constant-competitive online algorithm for the natural minimization version of correlation clustering.

Abstract

The standard online version of correlation clustering is prohibitively hard, as even randomized algorithms cannot achieve competitive ratio better than $Ω(n)$. Prior works bypass this lower bound by relaxing the online model through recourse, random arrival order, or seeding the algorithm with an offline sample of the underlying input. We instead ask whether additional structure in the input itself can overcome this lower bound. We study weighted correlation clustering under probability constraints, where $w^+_{uv}+w^-_{uv}=1$ for every $uv$ edge, and triangle inequality constraints, where the negative weights $w^-$ satisfy triangle inequality. While this version of correlation clustering is well-studied in the offline setting, we initiate its online study and give a deterministic online algorithm that maintains a clustering whose total weighted disagreement cost is within an $O(1)$ factor of the offline optimum, against adversarial arrival order. This is the first constant-competitive online algorithm for a natural minimization variant of correlation clustering in the fully online model, and shows that metric consistency on the edge weights separates tractable from intractable online instances.

Problem

• online correlation clustering
• metric weights
• triangle inequality
• competitive ratio
• weighted disagreement

Innovation

• online correlation clustering
• metric weights
• constant competitive ratio
• triangle inequality
• weighted disagreement