Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

The Complexity of Extending Storylines with Minimum Local Crossing Number

Alexander Dobler, Siddharth Gupta, Philipp Kindermann, Fabrizio Montecchiani, Martin Nöllenburg

TL;DR

It is proved that the problem is W[1]-hard parameterized by the number of inserted characters plus the maximum number $\sigma$ of active characters, in XP parameterized by $\sigma$ and in FPT parameterized by $\sigma+\chi$.

Abstract

Storyline layouts visualize temporal interactions by drawing each character as an $x$-monotone curve and enforcing that the participants of every meeting form a contiguous vertical group. We study a drawing extension variant in which a layout of a sub-storyline is fixed and has to be extended by inserting missing characters while preserving all meeting constraints. We minimize the local crossing number $χ$, i.e., the maximum number of crossings along any single character. We prove that the problem is W[1]-hard parameterized by the number $k$ of inserted characters plus the maximum number $σ$ of active characters, in XP parameterized by $σ$ and in FPT parameterized by $σ+χ$.

The Complexity of Extending Storylines with Minimum Local Crossing Number

TL;DR

It is proved that the problem is W[1]-hard parameterized by the number of inserted characters plus the maximum number of active characters, in XP parameterized by and in FPT parameterized by .

Abstract

Storyline layouts visualize temporal interactions by drawing each character as an -monotone curve and enforcing that the participants of every meeting form a contiguous vertical group. We study a drawing extension variant in which a layout of a sub-storyline is fixed and has to be extended by inserting missing characters while preserving all meeting constraints. We minimize the local crossing number , i.e., the maximum number of crossings along any single character. We prove that the problem is W[1]-hard parameterized by the number of inserted characters plus the maximum number of active characters, in XP parameterized by and in FPT parameterized by .
Paper Structure (4 sections, 5 theorems, 7 figures)

This paper contains 4 sections, 5 theorems, 7 figures.

Table of Contents

  1. Introduction
  2. Hardness of Local StoryLine Extension
  3. An - and -algorithm for Local StoryLine Extension
  4. Conclusions

Key Result

Theorem 2.1

LSLE parameterized by $k+\sigma$ is [1]-hard, even for $\mu=2$.

Figures (7)

  • Figure 1: Sketch of a storyline layout with six characters, seven meetings, and five time instants.
  • Figure 2: An $s$-saturator (left) and its schematic representation (right).
  • Figure 3: A $c$-channel (left) and its schematic representation (right).
  • Figure 4: An $x$-column gadget (left) and its schematic representation (right). The sparse channel has a light orange background, the dense channels are grey.
  • Figure 5: A schematic representation of $\Gamma'$.
  • ...and 2 more figures

Theorems & Definitions (5)

  • Theorem 2.1
  • Lemma 2.2
  • Lemma 2.3
  • Lemma 2.4
  • Theorem 3.1