Exact Approaches to Multilevel Vertical Orderings

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 multi-level vertical ordering (MLVO) problem falls into the class of quadratic ordering problems. It is conceptually related to the well-studied problem of multi-level crossing minimization (MLCM), but offers certain interesting novel properties: we not only have to consider the pure relative ordering of the nodes, but their final absolute ranks (i.e., positions) within the ordered levels. Furthermore, MLVO is a natural quadratic problem that does not only consist of multiple sequentially linked bilevel quadratic ordering problems, but is a genuine multi-level quadratic ordering problem. This allows us to describe the graphs’ structures more compactly and therefore obtain (near-)optimal, (well-)readable drawings of graphs too large for MLCM. We show (theoretically and experimentally) that these properties lead to the situation that approaches based on ILPs and QPs are inapplicable, even for small sparse graphs, while the SDP works surprisingly well in practice. This is in stark contrast to other ordering problems as, e.g., MLCM, where such graphs are typically solved more efficiently with ILPs. In this paper we present a motivation, mathematical models, strengthening constraints for ILPs and QPs, and an SDP relaxation for MLVO. We compare the relevant models from the polyhedral point of view, and conduct a series of experiments (including a comparison to MLCM) to showcase our SDP’s applicability. We conclude with sketching further applications in scheduling and ranking problems, where MLVO occurs apart from graph drawing.