RTD-Conjecture and Concept Classes Induced by Graphs

TL;DR

This paper confirms a conjecture relating recursive teaching dimension and VC-dimension for concept classes induced by graphs, specifically for classes of stars and connected sets, showing they are tightly bounded.

Contribution

It proves the RTD-VC conjecture for graph-induced concept classes, demonstrating the RTD is either equal to or just one less than the VC-dimension.

Findings

RTD equals VC-dimension or is less by 1 for classes induced by graphs

The conjecture holds for classes of stars and connected sets in graphs

Provides a tight bound between RTD and VC-dimension for these 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 $1$.