# The Computational Complexity of Positive Non-Clashing Teaching in Graphs

arXiv: https://arxiv.org/abs/2503.07665 · HTML: https://mlsanity.com/paper/2503.07665/

## Key facts

- Cited by 3 papers in the map (data through 14 Oct 2026).
- Builds on 18 of its 49 references that are in the map.
- First posted on arXiv 8 Mar 2025.

## Builds on (most cited first, 18 of 18)

- [An Overview of Machine Teaching](https://mlsanity.com/paper/1801.05927/) (2018)
- [Learning Mixtures of Linear Regressions with Nearly Optimal Complexity](https://mlsanity.com/paper/1802.07895/) (2018)
- [Machine Teaching for Inverse Reinforcement Learning: Algorithms and Applications](https://mlsanity.com/paper/1805.07687/) (2018)
- [Training Set Debugging Using Trusted Items](https://mlsanity.com/paper/1801.08019/) (2018)
- [Understanding the Role of Adaptivity in Machine Teaching: The Case of Version Space Learners](https://mlsanity.com/paper/1802.05190/) (2018)
- [Unlabeled sample compression schemes and corner peelings for ample and maximum classes](https://mlsanity.com/paper/1812.02099/) (2018)
- [Preference-Based Batch and Sequential Teaching: Towards a Unified View of Models](https://mlsanity.com/paper/1910.10944/) (2019)
- [Preference-based Teaching](https://mlsanity.com/paper/1702.02047/) (2017)
- [Sample compression schemes for balls in graphs](https://mlsanity.com/paper/2206.13254/) (2022)
- [Optimal Collusion-Free Teaching](https://mlsanity.com/paper/1903.04012/) (2019)
- [Non-Clashing Teaching Maps for Balls in Graphs](https://mlsanity.com/paper/2309.02876/) (2023)
- [On the Computational Complexity of Vertex Integrity and Component Order Connectivity](https://mlsanity.com/paper/1403.6331/) (2014)
- [A Parameterized Theory of PAC Learning](https://mlsanity.com/paper/2304.14058/) (2023)
- [Parameterized Complexity Dichotomy for Steiner Multicut](https://mlsanity.com/paper/1404.7006/) (2014)
- [Exploring the Gap Between Treedepth and Vertex Cover Through Vertex Integrity](https://mlsanity.com/paper/2101.09414/) (2021)
- [Extended MSO Model Checking via Small Vertex Integrity](https://mlsanity.com/paper/2202.08445/) (2022)
- [The Computational Complexity of Concise Hypersphere Classification](https://mlsanity.com/paper/2312.07103/) (2023)
- [Parameterized Vertex Integrity Revisited](https://mlsanity.com/paper/2402.09971/) (2024)

## Built on by (most cited first, 3 of 3)

- [The Parameterized Complexity of Computing the VC-Dimension](https://mlsanity.com/paper/2510.17451/) (2025)
- [Non-Clashing Teaching in Graphs: Algorithms, Complexity, and Bounds](https://mlsanity.com/paper/2602.00657/) (2026)
- [On Collusion-Free Teaching in Planar Graphs](https://mlsanity.com/paper/2610.18714/) (2026)

## Abstract (arXiv)

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.