Non-Clashing Teaching in Graphs: Algorithms, Complexity, and Bounds

TL;DR

We study the problem of (positive) non-clashing teaching for closed neighborhoods in graphs, showing that this graph-based concept class can encode any finite binary concept class. We derive ETH-based lower bounds for $N$-$NCTD$ and $N$-$NCTD^+$, and present algorithmic results including a $2^{ ext{O}}(|E(G)|)$-time solver for $N$-$NCTD^+$, along with FPT results by treedepth and by vertex cover. We also obtain combinatorial upper bounds for planar and unit-square graph classes. Together, these results advance understanding of teaching dimensions in graph-encoded concept classes and provide practical, structure-exploiting algorithms with potential broad impact on AI systems relying on algorithmic teaching.

Abstract

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.

Figures (8)

• Figure 1: Graph $G$ for a binary concept class $\mathcal{C}=\{\{1, 4\}, \{2, 5\}, \{2, 3, 5\}\}$ and $\mathcal{B} = \{N[C]\}_{C\in \mathcal{C}}$.
• Figure 2: Example of the graph $G$ constructed in the proof of Theorem \ref{['thm:genhard']} with $\varphi = (\overline{x_1}\vee {x_2}\vee \overline{x_3}) \wedge (\overline{x_2}\vee {x_3}\vee {x_4})$ in the 3-SAT instance. The two bold rectangles together form a clique. The edges going from vertices in one of the two large dashed rectangles to vertices in the other are omitted for legibility. Only the closed neighborhoods of filled vertices are in $\mathcal{B}$.
• Figure 3: Example of the graph $G$ constructed in the proof of Theorem \ref{['thm:poshard']} with $\varphi = (\overline{x_1}\vee \overline{x_2}\vee \overline{x_4}) \wedge (x_1\vee \overline{x_2}\vee {x_3})$ in the 3-SAT instance. Only the closed neighborhoods of filled vertices are in $\mathcal{B}$.
• Figure 4: (a) A graph $G$. (b) A treedepth decomposition $\mathcal{T}$ of $G$ witnessing $\mathtt{td}(G)\leq 4$.
• Figure 5: Cases 1, 2, and 3 in the forward direction of the proof of Lemma \ref{['lem:safe1']}. Recall that $y\notin N[v]$.

Theorems & Definitions (26)

• Lemma 1
• proof
• Theorem 2
• proof
• Theorem 3
• proof
• Theorem 4
• proof
• Theorem 5
• Lemma 6