Recently, Kirkpatrick et al. [ALT 2019] and Fallat et al. [JMLR 2023] introduced non-clashing teaching and showed it is the most efficient machine teaching model satisfying the Goldman-Mathias collusion-avoidance criterion. A teaching map $T$ for a concept class $\mathcal{C}$ assigns a (teaching) set $T(C)$ of examples to each concept $C \in \mathcal{C}$. A teaching map is non-clashing if no pair of concepts are consistent with the union of their teaching sets. The size of a non-clashing teaching map (NCTM) $T$ is the maximum size of a teaching set $T(C)$, $C \in \mathcal{C}$. The non-clashing teaching dimension NCTD$(\mathcal{C})$ of $\mathcal{C}$ is the minimum size of an NCTM for $\mathcal{C}$. NCTM$^+$ and NCTD$^+(\mathcal{C})$ are defined analogously, except the teacher may only use positive examples. We study NCTMs and NCTM$^+$s for the concept class $\mathcal{B}(G)$ consisting of all balls of a graph $G$. We show that the associated decision problem B-NCTD$^+$ for NCTD$^+$ is NP-complete in split, co-bipartite, and bipartite graphs. Surprisingly, we even prove that, unless the ETH fails, B-NCTD$^+$ does not admit an algorithm running in time $2^{2^{o(\text{vc})}}\cdot n^{O(1)}$, nor a kernelization algorithm outputting a kernel with $2^{o(\text{vc})}$ vertices, where vc is the vertex cover number of $G$. We complement these lower bounds with matching upper bounds. These are extremely rare results: it is only the second problem in NP to admit such a tight double-exponential lower bound parameterized by vc, and only one of very few problems to admit such an ETH-based conditional lower bound on the number of vertices in a kernel. For trees, interval graphs, cycles, and trees of cycles, we derive NCTM$^+$s or NCTMs for $\mathcal{B}(G)$ of size proportional to its VC-dimension, and for Gromov-hyperbolic graphs, we design an approximate NCTM$^+$ of size 2.

翻译：最近，Kirkpatrick等人[ALT 2019]与Fallat等人[JMLR 2023]提出了无冲突教学模型，并证明其是满足Goldman-Mathias防共谋准则的最高效机器学习教学模型。对于概念类$\mathcal{C}$，教学映射$T$为每个概念$C \in \mathcal{C}$分配一个（教学）示例集$T(C)$。若任意两个概念均不与其教学集之并集一致，则该教学映射称为无冲突的。无冲突教学映射（NCTM）$T$的规模定义为所有教学集$T(C)$（$C \in \mathcal{C}$）的最大基数。概念类$\mathcal{C}$的无冲突教学维度NCTD$(\mathcal{C})$是$\mathcal{C}$所有NCTM的最小规模。NCTM$^+$与NCTD$^+(\mathcal{C})$的定义类似，但教师仅能使用正例。本文研究图$G$中所有球体构成的概念类$\mathcal{B}(G)$的NCTM与NCTM$^+$。我们证明：对于分裂图、余二部图及二部图，NCTD$^+$对应的判定问题B-NCTD$^+$是NP完全问题。更令人惊讶的是，即使假设ETH（指数时间假设）成立，我们证明B-NCTD$^+$不存在$2^{2^{o(\text{vc})}}\cdot n^{O(1)}$时间复杂度的算法，也不存在能输出$2^{o(\text{vc})}$顶点规模核的核化算法，其中vc为图$G$的顶点覆盖数。我们通过匹配的上界结果对这些下界进行了补充。此类结果极为罕见：这是NP问题中第二个在vc参数下获得如此精确的双指数下界的问题，也是极少数能在核顶点数上获得基于ETH条件性下界的问题之一。对于树、区间图、环及环树结构，我们为$\mathcal{B}(G)$构建了规模与其VC维成比例的NCTM$^+$或NCTM；对于Gromov双曲图，我们设计了规模为2的近似NCTM$^+$。