Fixing Variables in Semidefinite Relaxations
SIAM Journal on Matrix Analysis and Applications
Two-Layer Planarization in Graph Drawing
ISAAC '98 Proceedings of the 9th International Symposium on Algorithms and Computation
Directed Graphs by Clan-Based Decomposition
GD '95 Proceedings of the Symposium on Graph Drawing
Which Aesthetic has the Greatest Effect on Human Understanding?
GD '97 Proceedings of the 5th International 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
Algorithms for multi-level graph planarity testing and layout
Theoretical Computer Science
A new lower bound for the single row facility layout problem
Discrete Applied Mathematics
Solving Max-Cut to optimality by intersecting semidefinite and polyhedral relaxations
Mathematical Programming: Series A and B
Exact Algorithms for the Quadratic Linear Ordering Problem
INFORMS Journal on Computing
Speeding up IP-based algorithms for constrained quadratic 0–1 optimization
Mathematical Programming: Series A and B - Series B - Special Issue: Combinatorial Optimization and Integer Programming
Decorous Lower Bounds for Minimum Linear Arrangement
INFORMS Journal on Computing
Optimal k-level planarization and crossing minimization
GD'10 Proceedings of the 18th international conference on Graph drawing
Improving layered graph layouts with edge bundling
GD'10 Proceedings of the 18th international conference on Graph drawing
An SDP approach to multi-level crossing minimization
Journal of Experimental Algorithmics (JEA)
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We present a semidefinite programming SDP approach for the problem of ordering vertices of a layered graph such that the edges of the graph are drawn as vertical as possible. This multilevel vertical ordering MLVO problem is a quadratic ordering problem and conceptually related to the well-studied problem of multilevel crossing minimization MLCM. In contrast to the latter, it can be formulated such that it does not merely consist of multiple sequentially linked bilevel quadratic ordering problems, but as a genuine multilevel problem with dense cost matrix. This allows us to describe the graphs' structures more compactly and therefore obtain solutions for graphs too large for MLCM in practice. In this paper we give motivation and mathematical models for MLVO. We formulate linear and quadratic programs, including some strengthening constraint classes, and an SDP relaxation for MLVO. We compare all approaches both theoretically and experimentally and show that MLVO's properties render linear and quadratic programming approaches inapplicable, even for small sparse graphs, while the SDP works surprisingly well in practice. This is in stark contrast to other ordering problems like MLCM, where such graphs are typically solved more efficiently with integer linear programs. Finally, we also compare our approach to related MLCM approaches.