Non-Clashing Teaching in Graphs: Algorithms, Complexity, and Bounds
Sujoy Bhore, Liana Khazaliya, Fionn Mc Inerney
TL;DR
This paper advances the understanding of non-clashing teaching in graphs by providing improved algorithms, complexity bounds, and combinatorial limits for the concept class of closed neighborhoods, broadening the scope beyond previous studies on balls in graphs.
Contribution
It introduces new FPT algorithms and stronger lower bounds for non-clashing teaching of closed neighborhoods, extending the complexity analysis to more general graph classes.
Findings
FPT algorithms for non-clashing teaching of closed neighborhoods
Stronger lower bounds on the problem's complexity
Upper bounds for wider classes of graphs
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…
Peer Reviews
Decision·ICLR 2026 Poster
The theoretical contribution is significant, as far as the reviewer can tell studying the immediate literature. And the math is correct, as far as the reviewer has checked (barring cited results). While the reviewer is not an expert in the immediate literature and cannot speak to how challenging it is to obtain such results, it is fair to say that the results are novel and comprehensive (spanning many different question, mostly solved to completion).
The paper is densely written and assumes some prior knowledge of the literature from the get-go (for instance, the jump from concept classes from balls in graphs in the introduction is a bit too sudden without explaining how a set of balls in G form a binary concept class, especially when graph quantities starts appearing in the bounds (e.g. line 67)). While this is expected from a theoretical paper, especially one that studies a rather niche problem, for the purpose of a machine learning conf
### Strengths The lower bound improvement for **N-NCTD** is significant, and the **FPT parameterization by vertex cover** for **N-NCTD** is highly relevant. I found the presentation style particularly effective, as several proofs are preceded by intuitive explanations and supported by neat, well-designed diagrams. This approach makes the paper engaging and easy to follow. I believe this style should be extended to other proofs as well, with additional illustrative examples where appropriate.
### Weaknesses The proofs and overall write-up are very specific and may primarily appeal to readers from the learning theory subcommunity. The paper is difficult to parse and, more importantly, to appreciate for readers without prior background in machine teaching or the concept of the non-clashing teaching dimension (NCTD). Although the proofs presented in Section 2 appear to be correct and I verified all of them rigorously, I found the graph construction to be very similar to the one used
1. Near-tight complexity results, improving lower bounds and adding new FPT regimes. 2. Careful comparison to balls in graphs, showing where closed neighborhoods help. 3. Combinatorial upper bounds tied to structural classes and VC-dimension observations.
1. Exposition can be heavy; more intuition and illustrative examples would aid accessibility. 2. Limited discussion of practical or empirical implications for machine teaching applications. 3. ETH tightness is strong theoretically, but could be complemented by empirical hardness studies.
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Taxonomy
TopicsMachine Learning and Algorithms · Complexity and Algorithms in Graphs · Advanced Graph Theory Research