This paper introduces the cluster linear program (LP) for correlation clustering, a powerful relaxation that unifies all previous LP formulations for the problem. The cluster LP is exponential in size but can be approximately solved in polynomial time for any ε > 0. This allows for the design of rounding algorithms without worrying about correlated rounding errors, which are handled uniformly in solving the relaxation. The paper demonstrates the power of the cluster LP by presenting a simple rounding algorithm and providing two analyses: one analytically proving a 1.49-approximation and another solving a factor-revealing SDP to show a 1.437-approximation. Both proofs introduce principled methods for analyzing the performance of the algorithm, resulting in a significantly improved approximation guarantee. The paper also proves an integrality gap of 4/3 for the cluster LP, showing that the 1.437-upper bound cannot be drastically improved. This result directly inspires an improved NP-hardness of approximation with a ratio of 24/23 ≈ 1.042. The cluster LP provides a unified framework for correlation clustering, and the paper shows how to solve it approximately in polynomial time, leading to improved approximation ratios for the problem.This paper introduces the cluster linear program (LP) for correlation clustering, a powerful relaxation that unifies all previous LP formulations for the problem. The cluster LP is exponential in size but can be approximately solved in polynomial time for any ε > 0. This allows for the design of rounding algorithms without worrying about correlated rounding errors, which are handled uniformly in solving the relaxation. The paper demonstrates the power of the cluster LP by presenting a simple rounding algorithm and providing two analyses: one analytically proving a 1.49-approximation and another solving a factor-revealing SDP to show a 1.437-approximation. Both proofs introduce principled methods for analyzing the performance of the algorithm, resulting in a significantly improved approximation guarantee. The paper also proves an integrality gap of 4/3 for the cluster LP, showing that the 1.437-upper bound cannot be drastically improved. This result directly inspires an improved NP-hardness of approximation with a ratio of 24/23 ≈ 1.042. The cluster LP provides a unified framework for correlation clustering, and the paper shows how to solve it approximately in polynomial time, leading to improved approximation ratios for the problem.