Liana Khazaliya

Sujoy Bhore, Liana Khazaliya, Fionn Mc Inerney

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.

Petr Hliněný, Liana Khazaliya

The crossing number of a graph is the minimum number of edge crossings that a graph can have when drawn in the plane. Determining this number, known as the Crossing Number problem, is a celebrated problem in combinatorial optimization. It has been known to be NP-complete since the 1980s, and already showing its fixed-parameter tractability when parameterized by the vertex cover number required fairly involved techniques. In this paper, we prove that computing the crossing number exactly remains NP-hard even for graphs of path-width 12 (and as a result, for simple graphs of path-width 13 and tree-width 9). These results highlight that, although both path- and tree-decompositions have been highly successful tools in many graph algorithm scenarios, general crossing number computation is unlikely (under P $\neq$ NP) to be successfully tackled using graph decompositions of bounded width -- a question that had remained a 'tantalizing open problem' [S. Cabello, Hardness of Approximation for Crossing Number, 2013] till now.

Robert Ganian, Liana Khazaliya, Fionn Mc Inerney et al.

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.