The Computational Complexity of Positive Non-Clashing Teaching in Graphs

2025-03-08

AI Summary

This paper investigates the computational complexity of the Positive Forward-Conflict-Free Teaching Dimension (PFD) under graph models—i.e., finding the smallest example set to unambiguously teach each concept to an intelligent learner. We first prove that PFD is NP-complete even for k = 2. We establish nearly tight bounds: an O*(2ⁿ) upper bound and a matching Ω*(2ⁿ) lower bound. Furthermore, we design a fixed-parameter tractable (FPT) algorithm parameterized by vertex integrity and rigorously show that PFD is not FPT when parameterized jointly by feedback vertex set size and pathwidth. By integrating graph-theoretic modeling (e.g., ball coverings, vertex integrity), classical complexity analysis, and parameterized algorithm design, we fully characterize the complexity landscape of PFD. Our results resolve several long-standing open problems in the literature and provide fundamental computability boundaries for machine teaching theory.

Technology Category

• Machine Learning: Probabilistic Circuits and Graphical Models
• Knowledge Representation and Reasoning: Computational Complexity of Reasoning
• Constraint Satisfaction and Optimization: Satisfiability Modulo Theories

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
• Web Mining and Content Analysis: Models for Web evolution

Abstract

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.

Problem

• NP-hardness of teaching dimension k=2 in graphs
• Fixed-parameter tractability by vertex integrity
• Lower bounds for feedback vertex number and pathwidth

Innovation

• NP-hardness proven for k=2 in graphs
• Fixed-parameter tractability by vertex integrity
• Lower bounds for feedback vertex number