AI 中文总结
研究在独特游戏猜想下簇删除问题的近似难度，证明其难以在2-ε因子内近似，通过归约得出√2-ε的近似难度，还给出31顶点图解答了簇编辑与坏三角横截最优值差异的开放问题。
AI 中文摘要
近期簇编辑（Cluster Editing）领域的突破性进展，促使研究人员尝试调整相关方法以获得优于2倍近似比的簇删除（Cluster Deletion）近似算法。我们在独特游戏猜想（Unique Games Conjecture）下排除了这种可能性：对于任意固定的ε>0，簇删除问题难以在2-ε的因子内近似，这与已知的2倍近似算法[Veldt等人，WWW 2018]结果一致。我们从顶点覆盖（Vertex Cover）出发的保近似归约，还意味着该问题难以在√2-ε的因子内近似。此外，我们证明在受限场景下可实现优于2倍的近似。最后，我们简要讨论了簇编辑与坏三角横截（Bad Triangle Transversal）的关系，特别给出了一个31顶点图G，其二者的最优值存在差异，解答了Adriaens和Tatti[ICML 2026]提出的开放问题。
英文摘要
Recent breakthroughs in Cluster Editing have motivated attempts to adapt these approaches to obtain better-than-$2$ approximations for Cluster Deletion. We rule out this possibility under the Unique Games Conjecture: Cluster Deletion is NP-hard to approximate within a factor of $2-ε$ for every fixed $ε>0$, matching the known $2$-approximation [Veldt et al., WWW 2018]. Our approximation-preserving reduction from Vertex Cover also implies NP-hardness of approximation within $\sqrt2-ε$. We also show that better-than-$2$ approximations are possible in restricted settings. We close the paper with a brief discussion of the relationship between Cluster Editing and Bad Triangle Transversal. In particular, we give a $31$-vertex graph~$G$ for which the two optimal values differ, answering an open question of Adriaens and Tatti [ICML 2026].
