The Computational Complexity of Positive Non-Clashing Teaching in Graphs

TL;DR

This work analyzes the computational complexity of the positive non-clashing teaching dimension for concept classes realized as balls in graphs. It establishes NP-hardness for the constant-dimensional case ($k=2$) even on restricted graph classes and derives near-tight exponential-time bounds under the Exponential Time Hypothesis, clarifying the landscape between lower and upper limits. The authors introduce a fixed-parameter tractable approach parameterized by vertex integrity $p$, employing kernelization and a novel canonical-structure analysis to obtain compact solutions. They also prove hardness results for combined structural parameters such as the feedback vertex number and pathwidth, thereby delineating the boundaries of tractability. Collectively, the results resolve several open questions from prior work and provide a near-complete understanding of the complexity of computing the positive non-clashing teaching dimension, with constructive implications for algorithm design in graph-based teaching models.

Abstract

We study the classical and parameterized complexity of computing the positive non-clashing teaching dimension of a set of concepts, that is, the smallest number of examples per concept required to successfully teach an intelligent learner under the considered, previously established model. For any class of concepts, it is known that this problem can be effortlessly transferred to the setting of balls in a graph G. We establish (1) the NP-hardness of the problem even when restricted to instances with positive non-clashing teaching dimension k=2 and where all balls in the graph are present, (2) near-tight running time upper and lower bounds for the problem on general graphs, (3) fixed-parameter tractability when parameterized by the vertex integrity of G, and (4) a lower bound excluding fixed-parameter tractability when parameterized by the feedback vertex number and pathwidth of G, even when combined with k. Our results provide a nearly complete understanding of the complexity landscape of computing the positive non-clashing teaching dimension and answer open questions from the literature.

Figures (2)

• Figure 1: Colorful edges denote paths of the depicted lengths. Two black curves separate the clique from the rest of the graph. Each dotted edge shows that a vertex of the clique is adjacent to all the vertices of the graph below the separating curves except those in the adjacent rectangle. The gray curve gives an intuition of which vertices of $P_x$, $P_y$, and $P_z$ are contained in $B_{r_c}(c)$ and $B_{r'_{c'}}(c')$.
• Figure :

Theorems & Definitions (43)

• Theorem 1
• Lemma 2
• proof
• Lemma 3
• proof
• Theorem 4
• Proposition 5
• proof
• Definition 1
• proof