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The Hierarchy of Saturating Matching Numbers

Hans U. Simon, Jan Arne Telle

TL;DR

This work investigates three machine-teaching-inspired matching problems through the lens of combinatorial parameters: the saturating matching number $SMN$, the antichain matching number $AMN$, and the greedy variants $GMN$, including their primed versions. It establishes a hierarchy of inequalities among these parameters, studies their monotonicity and additivity properties, and analyzes their behavior on the powerset class $P_n$ to derive tight asymptotic bounds. The central contributions include the extremal results for the $(k,n)$-family via $C_{k,n}$, a bound of $AMN'(C)$ by $GMN'(C)$, and extensive structural analysis (monotonicity, additivity, and sub-additivity) that clarifies how these costs interact under domain and class extensions and free combinations. The paper also highlights the tightness of the hierarchy with concrete separations and ends with an open question on the sub-additivity of $GMN$, signaling directions for future work. Overall, the results connect teaching-map cost measures to classical learning theory parameters (like $VCD$ and RTD) and provide a comprehensive map of how these combinatorial quantities relate and differ in representational settings relevant to teaching and learning.

Abstract

In this paper, we study three matching problems all of which came up quite recently in the field of machine teaching. The cost of a matching is defined in such a way that, for some formal model of teaching, it equals (or bounds) the number of labeled examples needed to solve a given teaching task. We show how the cost parameters associated with these problems depend on each other and how they are related to other well known combinatorial parameters (like, for instance, the VC-dimension).

The Hierarchy of Saturating Matching Numbers

TL;DR

This work investigates three machine-teaching-inspired matching problems through the lens of combinatorial parameters: the saturating matching number , the antichain matching number , and the greedy variants , including their primed versions. It establishes a hierarchy of inequalities among these parameters, studies their monotonicity and additivity properties, and analyzes their behavior on the powerset class to derive tight asymptotic bounds. The central contributions include the extremal results for the -family via , a bound of by , and extensive structural analysis (monotonicity, additivity, and sub-additivity) that clarifies how these costs interact under domain and class extensions and free combinations. The paper also highlights the tightness of the hierarchy with concrete separations and ends with an open question on the sub-additivity of , signaling directions for future work. Overall, the results connect teaching-map cost measures to classical learning theory parameters (like and RTD) and provide a comprehensive map of how these combinatorial quantities relate and differ in representational settings relevant to teaching and learning.

Abstract

In this paper, we study three matching problems all of which came up quite recently in the field of machine teaching. The cost of a matching is defined in such a way that, for some formal model of teaching, it equals (or bounds) the number of labeled examples needed to solve a given teaching task. We show how the cost parameters associated with these problems depend on each other and how they are related to other well known combinatorial parameters (like, for instance, the VC-dimension).

Paper Structure

This paper contains 9 sections, 7 theorems, 27 equations, 1 figure, 2 tables.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. A First Look at the SMN-GMN-AMN Hierarchy
  4. Bounding AMN$'$ by GMN$'$
  5. Monotonicity Considerations
  6. Evaluation of the Parameters on the Powerset
  7. Additivity Considerations
  8. A Final Look at the SMN-GMN-AMN Hierarchy
  9. Open Problem.

Key Result

Theorem 4.3

For each $1 \le d \le n$, the number of $C$-realizable samples of size $d$, with $C$ ranging over all concept classes in ${\mathcal{C}}_{k,n}$, is minimized by setting $C = C_{k,n}$.The main theorem in SHKT2024 is not stated in this form, but it is equivalent to what is claimed in Theorem th:hart-ge

Figures (1)

  • Figure 1: The parameter hierarchy: an arc $Y \rightarrow Z$ represents a relation $Y(C) \le Z(C)$ that is valid for each concept class $C$ and that is occasionally strict (for special choices of $C$).

Theorems & Definitions (39)

  • Remark 2.1
  • Example 4.1
  • Remark 4.2
  • Theorem 4.3: Main Theorem in SHKT2024
  • Corollary 4.4
  • Theorem 4.5
  • proof
  • Theorem 5.1
  • proof
  • Theorem 6.1
  • ...and 29 more