The cut polytope and the Boolean quadric polytope
Discrete Mathematics
A hierarchy of relaxation between the continuous and convex hull representations
SIAM Journal on Discrete Mathematics
Mathematical Programming: Series A and B
Facets for the cut cone II: clique-web inequalities
Mathematical Programming: Series A and B
An experimental comparison of four graph drawing algorithms
Computational Geometry: Theory and Applications
Fixing Variables in Semidefinite Relaxations
SIAM Journal on Matrix Analysis and Applications
A Technique for Drawing Directed Graphs
IEEE Transactions on Software Engineering
Directed Graphs by Clan-Based Decomposition
GD '95 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
Heuristics and meta-heuristics for 2-layer straight line crossing minimization
Discrete Applied Mathematics
Convex Optimization
Fixed parameter algorithms for one-sided crossing minimization revisited
Journal of Discrete Algorithms
Solving Max-Cut to optimality by intersecting semidefinite and polyhedral relaxations
Mathematical Programming: Series A and B
Layer-free upward crossing minimization
Journal of Experimental Algorithmics (JEA)
Exact Algorithms for the Quadratic Linear Ordering Problem
INFORMS Journal on Computing
Optimal k-level planarization and crossing minimization
GD'10 Proceedings of the 18th international conference on Graph drawing
GD'09 Proceedings of the 17th international conference on Graph Drawing
Exact Approaches to Multilevel Vertical Orderings
INFORMS Journal on Computing
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We present an approach based on semidefinite programs (SDP) to tackle the multi-level crossing minimization problem. We are given a layered graph (i.e., the graph's vertices are assigned to multiple parallel levels) and are asked for an ordering of the nodes on each level such that, when drawing the graph with straight lines, the resulting number of crossings is minimized. Solving this step is crucial in what is probably the most widely used graph drawing scheme, the Sugiyama framework. The problem has received a lot of attention in both the fields of heuristics and exact methods. For a long time, integer linear programming (ILP) approaches were the only exact algorithms applicable, at least for small graphs. Recently, SDP formulations for the special case of two levels were proposed and dominated the ILP for dense instances. In this article, we present a new SDP formulation for the general multi-level version that, for two levels, is even stronger than the aforementioned specialized SDP. As a by-product, we also obtain an SDP-based heuristic, which in practice always gives (near-)optimal solutions. We conduct a large set of experiments, both on randomized and on real-world instances, and compare our approach to a state-of-the-art ILP-based branch-and-cut implementation. The SDP clearly dominates for denser graphs, while the ILP approach is usually faster for sparse instances. However, even for such sparse graphs, the SDP solves more instances to optimality than the ILP. In fact, there is no single instance that the ILP solved that the SDP did not. Overall, our experiments reveal that, for sparse graphs, one should usually try to find an optimal solution with the ILP first. If this approach does not solve the instance to optimality within reasonable time, the SDP still has a good chance to do so. Being able to solve larger real-world instances than reported before, we are also able to evaluate heuristics for this problem. In this article, we do so for the traditional barycenter-heuristic (showing that it leaves a large gap to the true optimum) and the state-of-the-art upward-planarization method (showing that it is usually close to the optimum).