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Revisiting ILP Models for Exact Crossing Minimization in Storyline Drawings

Alexander Dobler, Michael Jünger, Paul J. Jünger, Julian Meffert, Petra Mutzel, Martin Nöllenburg

TL;DR

This paper revisits exact integer linear programming (ILP) approaches for this NP-hard problem and obtains exact solutions for an extended new benchmark set of larger and more complex instances than had been used before by enriching previous formulations with additional problem-specific insights and new heuristics.

Abstract

Storyline drawings are a popular visualization of interactions of a set of characters over time, e.g., to show participants of scenes in a book or movie. Characters are represented as $x$-monotone curves that converge vertically for interactions and diverge otherwise. Combinatorially, the task of computing storyline drawings reduces to finding a sequence of permutations of the character curves for the different time points, with the primary objective being crossing minimization of the induced character trajectories. In this paper, we revisit exact integer linear programming (ILP) approaches for this NP-hard problem. By enriching previous formulations with additional problem-specific insights and new heuristics, we obtain exact solutions for an extended new benchmark set of larger and more complex instances than had been used before. Our experiments show that our enriched formulations lead to better performing algorithms when compared to state-of-the-art modelling techniques. In particular, our best algorithms are on average 2.6-3.2 times faster than the state-of-the-art and succeed in solving complex instances that could not be solved before within the given time limit. Further, we show in an ablation study that our enrichment components contribute considerably to the performance of the new ILP formulation.

Revisiting ILP Models for Exact Crossing Minimization in Storyline Drawings

TL;DR

This paper revisits exact integer linear programming (ILP) approaches for this NP-hard problem and obtains exact solutions for an extended new benchmark set of larger and more complex instances than had been used before by enriching previous formulations with additional problem-specific insights and new heuristics.

Abstract

Storyline drawings are a popular visualization of interactions of a set of characters over time, e.g., to show participants of scenes in a book or movie. Characters are represented as -monotone curves that converge vertically for interactions and diverge otherwise. Combinatorially, the task of computing storyline drawings reduces to finding a sequence of permutations of the character curves for the different time points, with the primary objective being crossing minimization of the induced character trajectories. In this paper, we revisit exact integer linear programming (ILP) approaches for this NP-hard problem. By enriching previous formulations with additional problem-specific insights and new heuristics, we obtain exact solutions for an extended new benchmark set of larger and more complex instances than had been used before. Our experiments show that our enriched formulations lead to better performing algorithms when compared to state-of-the-art modelling techniques. In particular, our best algorithms are on average 2.6-3.2 times faster than the state-of-the-art and succeed in solving complex instances that could not be solved before within the given time limit. Further, we show in an ablation study that our enrichment components contribute considerably to the performance of the new ILP formulation.
Paper Structure (17 sections, 5 theorems, 12 equations, 2 figures)

This paper contains 17 sections, 5 theorems, 12 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Related Work.
  3. Contributions.
  4. Preliminaries
  5. Permutations.
  6. Problem input.
  7. Problem output.
  8. Standard Models for the Storyline Problem
  9. Structural Properties of Storyline Solutions
  10. Refining the ILP models
  11. The Propagated Linear Ordering Model (PLO)
  12. Symmetry Breaking Constraints
  13. Implementation
  14. Initial Heuristic
  15. Rounding and Local Improvement Heuristics
  16. ...and 2 more sections

Key Result

Lemma 6

Let $(T,\mathcal{C},\mathcal{I}, A)$ be an instance with a solution $S$. We can construct from $S$ a type-1-consistent solution $S'$ such that $\operatorname{{cr}}(S')\le \operatorname{{cr}}(S)$. If $S$ is type-2-consistent, so is $S'$.

Figures (2)

  • Figure 1: Storylines for the first Harry Potter movie. Interactions are shown as vertical gray bars.
  • Figure 2: Illustration of important notation throughout this paper with the aid of a storyline drawing depicting interactions $I_1$--$I_8$ of the characters $c_1$--$c_8$ over the time steps $t_1$--$t_7$

Theorems & Definitions (8)

  • Example 1
  • Definition 5: Type-1 consistency
  • Definition 6: Type-2 consistency
  • Lemma 6
  • Lemma 6
  • Corollary 7
  • Theorem 8
  • Theorem 9