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

2026-01-31

AI Summary

This work investigates the non-clashing teaching problem for concept classes defined by closed neighborhoods in graphs—a batch teaching model that satisfies anti-collusion criteria but suffers from high computational complexity and is only solvable on restricted graph classes. By leveraging parameterized complexity analysis, combinatorial graph-theoretic techniques, and reduction methods, we design the first fixed-parameter tractable (FPT) algorithm applicable to a significantly broader family of graphs. Our main contributions include an improved FPT algorithm for general graphs, tight combinatorial upper bounds, and proofs of strong inapproximability and W[1]-hardness results. These advances substantially deepen the understanding of both the algorithmic and complexity-theoretic landscape of this teaching problem.

Technology Category

• Machine Learning: Graph-based Machine Learning
• Constraint Satisfaction and Optimization: Other Foundations of Constraint Satisfaction
• Search and Optimization: Combinatorial Optimization

Application Category

• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs
• Semantics and Knowledge: Methods, algorithms and applications for the development of semantic models, knowledge graphs and other forms of structured data models with machine-interpretable semantics
• Responsible Web: Human-perceived consequences of algorithmic deployment on the web

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.

Problem

• non-clashing teaching
• closed neighborhoods
• graph classes
• machine teaching
• computational complexity

Innovation

• non-clashing teaching
• closed neighborhoods
• fixed-parameter tractability
• computational complexity
• graph classes