Star-Struck by Fixed Embeddings: Modern Crossing Number Heuristics

TL;DR

This paper evaluates crossing minimization heuristics using planarization and introduces a new mixed insertion method, highlighting the efficiency and quality benefits of fixed embedding star insertion over variable embedding methods.

Contribution

The paper provides a comprehensive experimental comparison of crossing minimization heuristics and introduces the mixed insertion method for improved performance.

Findings

Star insertion in fixed embeddings is faster than in variable embeddings.

Fixed embedding star insertion yields higher quality solutions.

Implementation details like non-simple crossing detection significantly impact heuristic performance.

Abstract

We present a thorough experimental evaluation of several crossing minimization heuristics that are based on the construction and iterative improvement of a planarization, i.e., a planar representation of a graph with crossings replaced by dummy vertices. The evaluated heuristics include variations and combinations of the well-known planarization method, the recently implemented star reinsertion method, and a new approach proposed herein: the mixed insertion method. Our experiments reveal the importance of several implementation details such as the detection of non-simple crossings (i.e., crossings between adjacent edges or multiple crossings between the same two edges). The most notable finding, however, is that the insertion of stars in a fixed embedding setting is not only significantly faster than the insertion of edges in a variable embedding setting, but also leads to solutions of higher quality.