The Complexity of Extending Storylines with Minimum Local Crossing Number

2026-03-09

AI Summary

This study addresses the extension problem in storyline layout: given an existing storyline with fixed character timelines and meeting continuity constraints, the goal is to insert a set of new characters so as to minimize the maximum number of crossings on any character’s curve—referred to as the local crossing number. Employing parameterized complexity theory, the work systematically analyzes the computational boundaries of this problem under various parameters. The main contributions include proving that the problem is W[1]-hard when parameterized by the sum of the number of inserted characters \(k\) and the number of simultaneously active characters \(\sigma\); showing it belongs to XP when parameterized solely by \(\sigma\); and establishing fixed-parameter tractability (FPT) when parameterized by the combined value of \(\sigma\) and the local crossing number \(\chi\). This last result provides the first clear delineation between tractability and intractability for optimizing local crossings in storyline extensions.

Technology Category

• Constraint Satisfaction and Optimization: Other Foundations of Constraint Satisfaction
• Knowledge Representation and Reasoning: Computational Complexity of Reasoning
• Search and Optimization: Combinatorial Optimization

Application Category

• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs
• Web Mining and Content Analysis: Models for Web evolution
• Systems and Infrastructure for Web, Mobile and WoT: Web performance, measurement, and characterization

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 $\chi$, 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 $\sigma$ of active characters, in XP parameterized by $\sigma$ and in FPT parameterized by $\sigma+\chi$.

Problem

• storyline layout
• drawing extension
• local crossing number
• temporal interactions
• meeting constraints

Innovation

• storyline layout
• local crossing number
• parameterized complexity
• drawing extension
• FPT