An open graph visualization system and its applications to software engineering
Software—Practice & Experience - Special issue on discrete algorithm engineering
Graph Drawing: Algorithms for the Visualization of Graphs
Graph Drawing: Algorithms for the Visualization of Graphs
An Alternative Method to Crossing Minimization on Hierarchical Graphs
GD '96 Proceedings of the Symposium on Graph Drawing
A Polyhedral Approach to the Multi-Layer Crossing Minimization Problem
GD '97 Proceedings of the 5th International Symposium on Graph Drawing
An SDP approach to multi-level crossing minimization
Journal of Experimental Algorithmics (JEA)
Exact Approaches to Multilevel Vertical Orderings
INFORMS Journal on Computing
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An important step in laying out hierarchical network diagrams is to order the nodes on each level. The usual approach is to minimize the number of edge crossings. This problem is NP-hard even for two layers when the first layer is fixed. Hence, in practice crossing minimization is performed using heuristics. Another suggested approach is to maximize the planar subgraph, i.e. find the least number of edges to delete to make the graph planar. Again this is performed using heuristics since minimal edge deletion for planarity is NP-hard.We show that using modern SAT and MIP solving approaches we can find optimal orderings for minimal crossing or minimal edge deletion for planarization on reasonably sized graphs. These exact approaches provide a benchmark for measuring quality of heuristic crossing minimization and planarization algorithms. Furthermore, we can straightforwardly extend our approach to minimize crossings followed by maximizing planar subgraph or vice versa; these hybrid approaches produce noticeably better layout then either crossing minimization or planarization alone.