The Computational Complexity of Positive Non-Clashing Teaching in Graphs
This resolves open questions in computational learning theory, offering a nearly complete complexity analysis for teaching dimension problems, though it is incremental as it builds on an established model.
The paper tackles the problem of computing the positive non-clashing teaching dimension for concept sets, showing it is NP-hard even for dimension k=2, provides near-tight runtime bounds, and establishes fixed-parameter tractability with vertex integrity but not with feedback vertex number or pathwidth.
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.