CKR Partitions and Lower Bounds for Constrained Correlation Clustering and Variants

2026-08-08

AI Summary

This study investigates the approximation limits of correlation clustering with must-link and cannot-link constraints and its variants, including Strong Triadic Closure with Minimum Weakness and Cluster Deletion. By reduction from Vertex Cover, it establishes that these three problems are Unique Games-hard to approximate within any constant factor below 2. The work introduces an innovative approach by incorporating the Calinescu–Karloff–Rabani (CKR) low-diameter decomposition into the linear programming relaxation framework for correlation clustering, replacing the conventional pivot-based strategy to elegantly handle edge constraints. Leveraging the standard semi-metric LP relaxation, CKR decomposition, and probabilistic analysis, the authors obtain a 3-approximation algorithm in expectation—the best-known approximation ratio to date—and further discuss the potential for surpassing this barrier.

Abstract

By using a simple textbook reduction from vertex cover, we show that the following three problems are all UG-hard to approximate with constant-factor smaller than two; minimum weakness strong triadic closure, cluster deletion and constrained correlation clustering. Additionally, we analyze the well-known low-diameter decomposition by Calinescu, Karloff and Raban applied to the standard LP relaxation semi-metric for constrained correlation clustering. As opposed to traditional pivot-based approaches, a CKR partition elegantly handles must-link and cannot-link constraints. It guarantees a 3-approximation in expectation, which matches the state-of-the-art approximation ratio by van Zuylen and Williamson. We conjecture that it in fact achieves a strictly better than 3-approximation, yet this remains an open problem.

Problem

• constrained correlation clustering
• cluster deletion
• strong triadic closure
• approximation hardness
• must-link/cannot-link constraints

Innovation

• CKR partition
• constrained correlation clustering
• low-diameter decomposition
• approximation algorithm
• must-link/cannot-link constraints