RTD-Conjecture and Concept Classes Induced by Graphs
Hans U. Simon
TL;DR
The paper investigates the RTD-conjecture, which seeks a universal bound of RTD by VC-dimension, and proves a strong form for two graph-induced concept families: stars and connected sets. By leveraging graph parameters such as the maximum degree $\Delta(G)$ and the leaves parameter $\ell(G)$, along with subset- and superset-based preference teaching, the authors establish tight bounds: for stars, $\Delta(G) \le \mathrm{RTD}(\mathcal{C}_{star}(G)) \le \mathrm{VCD}(\mathcal{C}_{star}(G)) \le \Delta(G)+1$, and for connected sets, $\ell(G) \le \mathrm{RTD}(\mathcal{C}_{con}(G)) \le \mathrm{VCD}(\mathcal{C}_{con}(G)) \le \ell(G)+1$. These results show that RTD and VCD differ by at most one in these natural graph-induced families, with several graphs illustrating all possible tightness scenarios. The findings strengthen the RTD-conjecture in concrete, structurally rich domains and illuminate connections to sample compression theory through teaching models and preferences. The work thus provides both tight theoretical bounds and methodological insights for extending RTD analysis to broader graph-based substructure classes.
Abstract
It is conjectured that the recursive teaching dimension of any finite concept class is upper-bounded by the VC-dimension of this class times a universal constant. In this paper, we confirm this conjecture for two rich families of concept classes where each class is induced by some graph $G$. For each $G$, we consider the class whose concepts represent stars in $G$ as well as the class whose concepts represent connected sets in $G$. We show that, for concept classes of this kind, the recursive teaching dimension either equals the VC-dimension or is less by $1$.