Online Correlation Clustering with Metric Weights
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.