Layered graph drawing

• Authors: Matthew Suderman
• Venue: Layered graph drawing
• Year: 2005

A layered graph drawing is a two-dimensional drawing of a combinatorial graph in which the vertices lie on a given set of horizontal lines. Such drawings are used in application domains such as software engineering, bioinformatics, and VLSI design. In addition to being layered, drawings in these applications may also satisfy other constraints, for example bounds on the number of edge crossings. The problems related to obtaining these drawings are almost always NP -hard, so, in this thesis, we investigate restricted versions of these problems in order to find efficient algorithmic solutions that can be used in practice. As a first very drastic restriction, we consider layered drawings that are planar. Even with this restriction, however, the resulting problems can still be NP -hard. In addition to proving one such hardness result, we do succeed in deriving efficient algorithms for two problems. In both cases, we correct previously published results that claimed extremely simple and efficient algorithmic solutions to these problems. Our solutions, though efficient as well, show that the truth about these problems is significantly more complex than the published results would suggest. We also study non-planar layered drawings, particularly drawings obtained by crossing minimization and minimum planarization. Though the corresponding problems are NP -hard, they become tractable when the value to be minimized is upper-bounded by a constant. This approach to obtaining tractable problems is formalized in a theory called parameterized complexity, and the resulting tractable problems and algorithmic solutions are said to be fixed-parameter tractable ( FPT ). Though relatively new, this theory has attracted a rapidly growing body of theoretical results. Indeed, we derive original FPT algorithms with the best-known asymptotic running times for planarization in two layer drawings. Because parameterized complexity is so new, little is known about its implications to the practice of graph drawing. Consequently, we have implemented a few FPT algorithms and compared them experimentally with previously implemented approaches, especially integer linear programming (ILP). Our experiments show that the performance of our FPT planarization algorithms are competitive with current ILP algorithms, but that, for crossing minimization, current ILP algorithms remain the clear winners.