AI 中文总结
该研究通过归约证明三类相关聚类问题是UG困难的，分析CKR划分算法对约束的处理能力，其期望近似比达3，推测可实现更优近似比，该推测仍待验证。
AI 中文摘要
通过从顶点覆盖问题进行简单的教材式归约，我们证明以下三个问题均是唯一游戏（Unique Game, UG）困难的，无法以小于2的常数因子近似：最小强度强三元闭包问题、簇删除问题以及约束相关聚类问题。此外，我们分析了Calinescu、Karloff和Raban提出的著名低直径分解算法，将其应用于约束相关聚类的标准线性规划松弛半度量问题。与传统的基于枢轴的方法不同，CKR划分可巧妙处理必须链接（must-link）和不能链接（cannot-link）约束，它保证了期望意义下的3近似比，与van Zuylen和Williamson提出的当前最优近似比一致。我们推测其实际上可实现优于3的近似比，但这仍是一个未解决的问题。
英文摘要
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 original 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.
CommentsIndependent and concurrent of our work, both Cao and Xu~\cite{cao2026cluster} and Azizeddin et al.~\cite{azizeddin2026constrained} obtained the same hardness results for CD and CCC
