Solving the Correlation Cluster LP in Sublinear Time

TL;DR

The paper tackles Correlation Clustering by leveraging the cluster LP, which, despite its exponential size, can be approximated efficiently. It introduces a sublinear-time method to compute a near-optimal LP solution and extends this to a rounding procedure that achieves a $(1.485+\varepsilon)$-approximation for Correlation Clustering, matching state-of-the-art polynomial-time results with significantly reduced running time. Central to the approach are (i) a multiplicative-weights-based solver for a covering reformulation of the cluster LP, (ii) a preclustering step that yields structured atoms and admissible edges to guide clustering, (iii) a partial clustering strategy that iteratively extracts small-ratio clusters to cover a constant fraction of mass, and (iv) scalable MPC and sublinear implementations of both LP solving and rounding. Together, these techniques bridge the gap between high-accuracy approximation algorithms and fast, scalable clustering, enabling practical applications in large-scale data analysis where only sublinear or near-linear runtimes are feasible.

Abstract

Correlation Clustering is a fundamental and widely-studied problem in unsupervised learning and data mining. The input is a graph and the goal is to construct a clustering minimizing the number of inter-cluster edges plus the number of missing intra-cluster edges. CCL+24 introduced the cluster LP for Correlation Clustering, which they argued captures the problem much more succinctly than previous linear programming formulations. However, the cluster LP has exponential size, with a variable for every possible set of vertices in the input graph. Nevertheless, CCL+24 showed how to find a feasible solution for the cluster LP in time $O(n^{\text{poly}(1/ε)})$ with objective value at most $(1+ε)$ times the value of an optimal solution for the respective Correlation Clustering instance. Furthermore, they showed how to round a solution to the cluster LP, yielding a $(1.485+ε)$-approximation algorithm for the Correlation Clustering problem. The main technical result of this paper is a new approach to find a feasible solution for the cluster LP with objective value at most $(1+ε)$ of the optimum in time $\widetilde O(2^{\text{poly}(1/ε)} n)$, where $n$ is the number of vertices in the graph. We also show how to implement the rounding within the same time bounds, thus achieving a fast $(1.485+ε)$-approximation algorithm for the Correlation Clustering problem. This bridges the gap between state-of-the-art methods for approximating Correlation Clustering and the recent focus on fast algorithms.

Figures (1)

• Figure 1: Illustration of the sets $\hat{T}_i$, $C^*_i$, and $Q_i$. The rectangle represents $D(r)$, divided into $\eta$ parts. The red region denotes $\hat{T}_i$, containing all vertices already added to $\hat{T}$. The set $C^*_i$ includes both the red and blue regions. In $D^{i+1}_r$, the algorithm attempts to include as many vertices as possible in $C^*_i$; the yellow region represents the newly added vertices in $\hat{T}$. Claim \ref{['claim:invariant-4.2']} states that the yellow and blue regions have significant overlap.

Theorems & Definitions (108)

• Theorem 1: Efficient \ref{['LP:clusterlp']}
• Theorem 2: Efficient Rounding Algorithm
• Corollary 3
• Definition 4: Preclustering
• Definition 5: $\varepsilon$-similar Preclustering
• Definition 6: $\varepsilon$-large Cluster
• Theorem 7: Preclustering Procedures cohen2024combinatorial
• Lemma 8
• proof
• Lemma 9