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 typical approximate optimization tasks, we propose the cluster LP as a strong linear program for Correlation Clustering. We demonstrate the power of the cluster LP by presenting new rounding algorithms, and providing two analyses, one analytically proving a 1.56-approximation and the other solving a factor-revealing SDP to show a 1.485-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 1.485-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.

翻译：在Bansal、Blum和Chawla（FOCS 2002）提出的经典相关性聚类问题中，输入是一个完全图，其边被标记为$+$或$-$，目标是找到一个顶点划分，使得跨部分的+边之和与部分内的-边之和最小化。近年来，Chawla、Makarychev、Schramm和Yaroslavtsev（STOC 2015）通过对标准线性规划进行近乎最优的舍入，给出了2.06近似算法；而Cohen-Addad、Lee、Li和Newman（FOCS 2022, 2023）最终突破了该线性规划的整数性间隙2，为该问题提供了$1.73$近似算法。为了构建一个类似于典型近似优化任务的简单统一框架，我们提出簇线性规划作为相关性聚类的强线性规划。我们通过提出新的舍入算法展示了簇线性规划的威力，并提供了两种分析：一种通过解析方法证明了1.56近似比，另一种通过求解因子揭示半定规划显示了1.485近似比。两种证明均引入了分析算法性能的原则性方法，从而显著改进了近似保证。最后，我们证明了簇线性规划的整数性间隙为$4/3$，表明我们的1.485上界无法大幅改进。我们的间隙实例直接启发了一个改进的NP难近似比$24/23 \\approx 1.042$；此前未发现明确的硬度比。