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Graph drawing applications in combinatorial theory of maturity models

Špela Kajzer, Alexander Dobler, Janja Jerebic, Martin Nöllenburg, Joachim Orthaber, Drago Bokal

TL;DR

This paper introduces tiled graphs as models of learning and maturing processes, and shows how tiled graphs can combine graphs of learning spaces or antimatroids and maturity models to yield models of learning processes.

Abstract

In this paper, we introduce tiled graphs as models of learning and maturing processes. We show how tiled graphs can combine graphs of learning spaces or antimatroids (partial hypercubes) and maturity models (total orders) to yield models of learning processes. For the visualization of these processes it is a natural approach to aim for certain optimal drawings. We show for most of the more detailed models that the drawing problems resulting from them are NP-complete. The terse model of a maturing process that ignores the details of learning, however, results in a polynomially solvable graph drawing problem. In addition, this model provides insight into the process by ordering the subjects at each test of their maturity. We investigate extremal and random instances of this problem, and provide exact results and bounds on their optimal crossing number. Graph-theoretic models offer two approaches to the design of optimal maturity models given observed data: (1) minimizing intra-subject inconsistencies, which manifest as regressions of subjects, is modeled as the well-known feedback arc set problem. We study the alternative of (2) finding a maturity model by minimizing the inter-subject inconsistencies, which manifest as crossings in the respective drawing. We show this to be NP-complete.

Graph drawing applications in combinatorial theory of maturity models

TL;DR

This paper introduces tiled graphs as models of learning and maturing processes, and shows how tiled graphs can combine graphs of learning spaces or antimatroids and maturity models to yield models of learning processes.

Abstract

In this paper, we introduce tiled graphs as models of learning and maturing processes. We show how tiled graphs can combine graphs of learning spaces or antimatroids (partial hypercubes) and maturity models (total orders) to yield models of learning processes. For the visualization of these processes it is a natural approach to aim for certain optimal drawings. We show for most of the more detailed models that the drawing problems resulting from them are NP-complete. The terse model of a maturing process that ignores the details of learning, however, results in a polynomially solvable graph drawing problem. In addition, this model provides insight into the process by ordering the subjects at each test of their maturity. We investigate extremal and random instances of this problem, and provide exact results and bounds on their optimal crossing number. Graph-theoretic models offer two approaches to the design of optimal maturity models given observed data: (1) minimizing intra-subject inconsistencies, which manifest as regressions of subjects, is modeled as the well-known feedback arc set problem. We study the alternative of (2) finding a maturity model by minimizing the inter-subject inconsistencies, which manifest as crossings in the respective drawing. We show this to be NP-complete.
Paper Structure (12 sections, 16 theorems, 41 equations, 12 figures, 1 algorithm)

This paper contains 12 sections, 16 theorems, 41 equations, 12 figures, 1 algorithm.

Table of Contents

  1. Introduction
  2. Learning graphs and their applications
  3. -completeness of tile crossing number
  4. Optimal ordinal panel data drawings can be obtained in polynomial time
  5. Extremal instances
  6. Extremal general instances
  7. Extremal consistent instances
  8. Random instances
  9. Optimal Maturity Models
  10. -completeness.
  11. Integer program formulation.
  12. Conclusion

Key Result

Theorem 3.2

Given $k\in\mathbb{N}$, it is -complete to decide whether a possibilistic learning tile has an ordinal panel drawing with at most $k$ crossings, even for a single subject, a single category, and a single tile.

Figures (12)

  • Figure 1: Example of a graph drawing of a learning space.
  • Figure 2: Example of a tile drawing.
  • Figure 3: Two non-trivial compatible tiles and their join.
  • Figure 4: Graph drawing of a learning space together with a ranking function assigning four maturity levels to knowledge states.
  • Figure 5: Total learning tile
  • ...and 7 more figures

Theorems & Definitions (44)

  • Definition 2.1
  • Definition 2.2: pinontoan2003crossing
  • Definition 2.3
  • Definition 2.4
  • Definition 2.5
  • Definition 2.6
  • Definition 2.7
  • Definition 2.8
  • Theorem 3.2
  • proof
  • ...and 34 more