Recently, Kirkpatrick et al. [ALT 2019] and Fallat et al. [JMLR 2023] introduced non-clashing teaching and showed it to be the most efficient machine teaching model satisfying the benchmark for collusion-avoidance set by Goldman and Mathias. A teaching map $T$ for a concept class $\cal{C}$ assigns a (teaching) set $T(C)$ of examples to each concept $C \in \cal{C}$. 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 $T(C)$, $C \in \cal{C}$. The non-clashing teaching dimension NCTD$(\cal{C})$ of $\cal{C}$ is the minimum size of an NCTM for $\cal{C}$. NCTM$^+$ and NCTD$^+(\cal{C})$ 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 {\sc B-NCTD$^+$} for NCTD$^+$ is NP-complete in split, co-bipartite, and bipartite graphs. Surprisingly, we even prove that, unless the ETH fails, {\sc B-NCTD$^+$} does not admit an algorithm running in time $2^{2^{o(vc)}}\cdot n^{O(1)}$, nor a kernelization algorithm outputting a kernel with $2^{o(vc)}$ vertices, where vc is the vertex cover number of $G$. These are extremely rare results: it is only the second (fourth, resp.) problem in NP to admit a double-exponential lower bound parameterized by vc (treewidth, resp.), and only one of very few problems to admit an ETH-based conditional lower bound on the number of vertices in a kernel. We complement these lower bounds with matching upper bounds. 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. For Gromov-hyperbolic graphs, we design an approximate NCTM$^+$ for $\mathcal{B}(G)$ of size 2.

翻译：近期，Kirkpatrick等[ALT 2019]和Fallat等[JMLR 2023]引入了非冲突教学，并证明其是在Goldman与Mathias设定的防合谋基准下最高效的机器教学模型。教学映射$T$为概念类$\cal{C}$中的每个概念$C$分配一组教学示例$T(C)$。当任意两个概念都不与它们教学集的并集一致时，该教学映射称为非冲突的。非冲突教学映射(NCTM) $T$的大小定义为所有$T(C)$($C \in \cal{C}$)的最大尺寸。概念类$\cal{C}$的非冲突教学维度NCTD$(\cal{C})$是$\cal{C}$的NCTM的最小尺寸。类似地可定义NCTM$^+$与NCTD$^+(\cal{C})$，区别在于教师仅使用正例。我们研究图$G$的全体球构成的概念类$\mathcal{B}(G)$的NCTM与NCTM$^+$。结果表明，关于NCTD$^+$的判定问题{\sc B-NCTD$^+$}在分裂图、余二部图和二部图中是NP完全的。令人惊讶的是，我们进一步证明：除非ETH失效，否则{\sc B-NCTD$^+$}无法在$2^{2^{o(vc)}}\cdot n^{O(1)}$时间内求解，也无法输出包含$2^{o(vc)}$个顶点的核化算法，其中vc是$G$的顶点覆盖数。这些结果极为罕见：它们是NP中第二个（第四个）以vc（树宽）为参数呈现双指数下界的问题，也是极少数能对核的顶点数给出基于ETH条件性下界的问题之一。我们通过匹配上界补充这些下界。对于树、区间图、环以及环的树，我们推导出尺寸与$\mathcal{B}(G)$的VC维成正比的NCTM$^+$或NCTM。对于格罗莫夫双曲图，我们设计了尺寸为2的$\mathcal{B}(G)$近似NCTM$^+$。