Kirkpatrick et al. [ALT 2019] and Fallat et al. [JMLR 2023] introduced non-clashing teaching and proved that it is the most efficient batch machine teaching model satisfying the collusion-avoidance benchmark established in the seminal work of Goldman and Mathias [COLT 1993]. Recently, (positive) non-clashing teaching was thoroughly studied for balls in graphs, yielding numerous algorithmic and combinatorial results. In particular, Chalopin et al. [COLT 2024] and Ganian et al. [ICLR 2025] gave an almost complete picture of the complexity landscape of the positive variant, showing that it is tractable only for restricted graph classes due to the non-trivial nature of the problem and concept class. In this work, we consider (positive) non-clashing teaching for closed neighborhoods in graphs. This concept class is not only extensively studied in various related contexts, but it also exhibits broad generality, as any finite binary concept class can be equivalently represented by a set of closed neighborhoods in a graph. In comparison to the works on balls in graphs, we provide improved algorithmic results, notably including FPT algorithms for more general classes of parameters, and we complement these results by deriving stronger lower bounds. Lastly, we obtain combinatorial upper bounds for wider classes of graphs.

翻译：Kirkpatrick等[ALT 2019]与Fallat等[JMLR 2023]引入了非冲突教学，并证明了其是满足Goldman与Mathias[COLT 1993]开创性工作中建立的避共谋基准的最优批量机器教学模式。近期，学界对图中球的（正）非冲突教学进行了深入研究，获得了大量算法与组合结果。具体而言，Chalopin等[COLT 2024]与Ganian等[ICLR 2025]给出了正变体复杂度景观的近乎完整的图景，表明由于问题与概念类的非平凡性质，该变体仅对受限图类可解。本文研究图中闭邻域上的（正）非冲突教学。该概念类不仅在多种相关语境中被广泛研究，且具有广泛普适性——任何有限二元概念类均可通过图中的闭邻域集合等价表示。与图中球的相关研究相比，我们给出了改进的算法结果，特别包括针对更一般参数类的FPT算法，并通过推导更强的下界对这些结果进行补充。最后，我们获得了更广泛图类的组合上界。