Non-Clashing Teaching Maps for Balls in Graphs

TL;DR

This work studies non-clashing teaching maps for balls in graphs, focusing on the concept class $\mathcal{B}(G)$ and two teaching dimensions $\text{NCTD}$ and $\text{NCTD}^+$. It proves strong complexity results, including NP-completeness and ETH-based lower bounds for parameterized versions, while also providing tight upper bounds and constructive maps for several graph families. The paper shows that for trees, interval graphs, cycles, and trees of cycles, NCTMs can be achieved with sizes tied to VC-dimension, and it provides a 2-size approximate NCTM^+ for $\delta$-hyperbolic graphs. The results offer both theoretical insights into the limits of batch teaching with collusion-avoidance and practical constructions of compact teaching maps in rich graph classes, with implications for learning theory and graph-based concept representations.

Abstract

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.

Figures (5)

• Figure 1: Graph $G$ in the proof of Thm. \ref{['thm:vc-lower-bound']} (left) and its sets $A^{\alpha}$, $V^{\alpha}$, $C$ (right). For clarity, we omit $C^{\gamma},V^{\gamma},V^{\gamma,*},A^{\gamma}$, $u'_{3M+1}$. In $\phi$, $x^{\alpha}_1$ appears as a positive literal in $C_1$, and $x^{\alpha}_2$ as a negative literal in $C_2$. Red and blue edges are according to $\textsf{set-rep}$ (in a complementary way).
• Figure 2: $Z(x,u,v)$, $Z^u(x,u,v)$, $Z^v(x,u,v)$, $s$, and $t$.
• Figure 3: The split graph $G$ constructed in the proof of Theorem \ref{['thm:split']}. Vertices contained in a rectangle form a clique. An edge between two ellipses indicates that each vertex in one ellipse is adjacent to each vertex in the other. Dashed lines highlight some non-existing edges. In this example, $2,3\in S_1$ (but $1,n\notin S_1$) and $1,3\in S_m$ (but $2,n\notin S_m$) in $\phi$.
• Figure 4: The co-bipartite graph $G$ constructed in the proof of Theorem \ref{['thm:split']}. An edge between a vertex and an ellipse indicates that vertex is adjacent to each vertex in the ellipse.
• Figure 5: The bipartite graph $G$ constructed in the proof of Theorem \ref{['thm:split']}. An edge between a vertex and an ellipse indicates that vertex is adjacent to each vertex in the ellipse. See the caption of Figure \ref{['fig:split']} for more details.

Theorems & Definitions (47)

• Example 1
• Example 2
• Theorem 3
• Proposition 3
• Theorem 4
• Proposition 5
• Lemma 5
• Lemma 5
• Theorem 6
• Lemma 6