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

TL;DR

This work addresses online correlation clustering by introducing an online-with-a-sample (AOS) model that enables simultaneous approximation for all ℓ_p-norm objectives. The authors design a single online algorithm that, using an ε-fraction sample, achieves competitive guarantees of the form O((1/ε^8) log^4 n) for all finite p, O((1/ε^6) log n) for the ℓ_∞-norm, and O(1/ε^6) in expectation for the ℓ_1-norm, relative to the offline OPT_p. The technical core combines sampling-based semi-metric estimation (via d̄ and the adjusted d̃) with a online-adapted CGS rounding, a pre-clustering phase, and a Pivot-based phase, all analyzed under a high-probability good event B^c. The paper also proves a fundamental RO-hardness separation for the ℓ_∞-norm and near-tight lower bounds in the AOS model for ℓ_1 and ℓ_∞, motivating the Beyond-Worst-Case model. Overall, the results translate the offline all-norms guarantee into the online setting, revealing the power of AOS with sample-driven surrogate metrics for multi-objective clustering problems.

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.

Figures (9)

• Figure 1: Overview of lemmas for the analysis of Algorithm \ref{['alg: main-alg']} for finite $p$. Solid edges are disagreements, and dashed ovals are clusters. Vertices are partitioned into three sets based on whether or not they are eligible and pre-clustered, eligible and not pre-clustered, or ineligible. Charging the cost of a disagreement then depends on which set its endpoints belong to, its sign, and potentially which endpoint was higher with respect to the partial ordering $\succ$. Edges are partitioned by color, with edge types of the same color bounded by the correspondingly colored lemma. Note Lemmas \ref{['lem: type1b-lp']} and \ref{['lem: type1b-lp-backwards']} correspond to the same $uv$ pair, but which lemma is relevant depends on (from the perspective of $u$'s disagreements) whether $u \succ v$ or $v \succ u$.
• Figure 2: Bounding $E_{1b}$ in the proof of Lemma \ref{['lem: type1b-lp']}, where $v \in N_u^+$ with $v \succ u$ and $\tilde{d}_{uv} \leq t$. We charge the disagreements between $v \in E_1(u)$ and $u$ to the purple nodes in $\text{Ball}^{S_b}_{\tilde{d}}(s^*(v),r)$.
• Figure 3: Bounding $E_{1b}$ in the proof of Lemma \ref{['lem: type1b-lp-backwards']}. We show all the purple nodes are sufficiently far from $u$. Then, we will show that the number of orange nodes, $|E_1(u)|$, is roughly equal to the number of purple nodes, $\text{Ball}_{\tilde{d}}(s^*(u),r)$, so we can charge the disagreements $uv$ for $v \in E_1(u)$ to the cost of $\tilde{d}$ on edges $uw$ for $w \in \text{Ball}_{\tilde{d}}(s^*(u),r)$.
• Figure 4: An overview of the cost analysis of Algorithm \ref{['alg: main-alg']} for $p=1$. As in Figure \ref{['fig:summary-ellp']}, solid edges are disagreements and dashed ovals are clusters, and different types of disagreements are color coded with the corresponding lemmas where we handle their charging arguments.
• Figure 5: Bounding $E_{1b}$ in the proof of Lemma \ref{['lem: type1b']}. The orange points $v \in E_1(u)$ are clustered before $u$, so edges $uv$ are disagreements. To bound the cost of the orange points, it suffices to show that there are more purple points in expectation, and that the cost of the purple points can be charged to the fractional cost $\tilde{D}_0(u)$ of $u$.

Theorems & Definitions (173)

• Theorem 1
• Theorem 2
• Theorem 3
• Definition 1
• Definition 2: Correlation metric
• Definition 3: Adjusted correlation metric
• Definition 4: Estimated correlation metric
• Definition 5: Estimated adjusted correlation metric
• Definition 6
• Lemma 1