Competitive Random-Order Correlation k-Clustering

2026-09-28

AI Summary

This study addresses online algorithms for the NP-hard correlation k-clustering problem under the random-order model. Motivated by the classical Pivot algorithm, we design an online strategy tailored to the random vertex arrival setting and employ theoretical analysis techniques specific to the random-order framework. Our primary contributions are threefold. First, we establish the inaugural Ω(log k) lower bound on the competitive ratio. Second, we derive a polylogarithmic upper bound, thereby closing a critical theoretical gap for general values of k in the online setting. Third, we achieve a polynomial-time constant-factor approximation that refines the online competitive ratio to the polylogarithmic level, effectively overcoming the limitations inherent in traditional offline approaches.

Abstract

Correlation clustering has been extensively studied over the last two decades in many different computational models owing to its wide-ranging practical applications. The goal is to compute a partition of the vertex set such that the total number of disagreements, i.e. the sum of edges between clusters, and non-edges within clusters, is minimized. In this paper, we study correlation $k$-clustering, in which the total number of clusters is restricted to $k$. Correlation $k$-clustering is NP-hard, and while previous work has presented a polynomial time approximation scheme for constant $k$, there are no known results for general $k$. Our first result is a polynomial-time constant-factor approximation algorithm for correlation $k$-clustering for general $k$. The main focus of this work is in the more challenging online setting. Noting that the best competitive ratio under adversarial arrivals is known to be $\Omega(n)$, we concentrate on the well-studied random-order model, where vertices arrive in random order and on arrival of a vertex, edges to its earlier-arrived neighbors are revealed. We prove a surprising lower bound of $\Omega(\log k)$-competitiveness for any online algorithm, which can be extended to $\Omega(\log n)$ when $k = \text{poly}(n)$. Finally, the main result of this paper is a polylogarithmic upper bound on the competitive ratio for correlation $k$-clustering, using an algorithm inspired by the classic Pivot algorithm for correlation clustering.

Problem

• Correlation k-clustering
• Online algorithms
• Random-order model
• Competitive ratio
• Approximation algorithms

Innovation

• Correlation k-clustering
• Approximation algorithm
• Random-order model
• Competitive ratio
• Pivot algorithm