  In the classic Correlation Clustering problem introduced by Bansal, Blum, and
Chawla~(FOCS 2002), the input is a complete graph where edges are labeled
either $+$ or $-$, and the goal is to find a partition of the vertices that
minimizes the sum of the +edges across parts plus the sum of the -edges within
parts. In recent years, Chawla, Makarychev, Schramm and Yaroslavtsev~(STOC
2015) gave a 2.06-approximation by providing a near-optimal rounding of the
standard LP, and Cohen-Addad, Lee, Li, and Newman~(FOCS 2022, 2023) finally
bypassed the integrality gap of 2 for this LP giving a $1.73$-approximation for
the problem.
  In order to create a simple and unified framework for Correlation Clustering
similar to those for {\em typical} approximate optimization tasks, we propose
the {\em cluster LP} as a strong linear program that might tightly capture the
approximability of Correlation Clustering. It unifies all the previous
relaxations for the problem.
  We demonstrate the power of the cluster LP by presenting a simple rounding
algorithm, and providing two analyses, one analytically proving a
1.49-approximation and the other solving a factor-revealing SDP to show a
1.437-approximation. Both proofs introduce principled methods by which to
analyze the performance of the algorithm, resulting in a significantly improved
approximation guarantee.
  Finally, we prove an integrality gap of $4/3$ for the cluster LP, showing our
1.437-upper bound cannot be drastically improved. Our gap instance directly
inspires an improved NP-hardness of approximation with a ratio $24/23 \approx
1.042$; no explicit hardness ratio was known before.
