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等人[ALT 2019]与Fallat等人[JMLR 2023]提出了非冲突教学模型，并证明其是满足Goldman和Mathias开创性工作[COLT 1993]所建立防串通基准的最高效批量机器教学模型。近年来，针对图中球集的正例非冲突教学得到了深入研究，产生了大量算法与组合学结果。特别地，Chalopin等人[COLT 2024]和Ganian等人[ICLR 2025]几乎完整描绘了正例变体的复杂度图景，表明由于问题与概念类的非平凡特性，该问题仅在受限图类中可高效求解。本文研究图中闭邻域的正例非冲突教学。该概念类不仅在诸多相关研究中被广泛探讨，更具有普遍性——任何有限二元概念类均可等价表示为图中闭邻域的集合。相较于图中球集的研究，我们提出了改进的算法结果，特别是针对更广泛参数类的FPT算法，并通过推导更强的下界对这些结果进行补充。最后，我们为更广泛的图类建立了组合上界。