Exact Approaches to Multilevel Vertical Orderings

Abstract

W e 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.

A semidefinite optimization-based approach for global optimization of multi-row facility layout

“…We therefore adapt an approach originally suggested in Fischer et al [21] that was successful for the max-cut problem [48] and several ordering problems [14,33].…”

Multi-level Verticality Optimization: Concept, Strategies, and Drawing Scheme

“…Therefore, it is beyond the scope of this paper to give in-depth details of inner working of the rather complex exact algorithms to tackle the problem. For such a discussion see [4], where we consider ILP-, QP-, and SDP-based algorithms to tackle the base problem of MLVO. Although graph drawing is the main (and most developed) application area, MLVO can also be interesting in other, very diverse, application fields like scheduling and multiple ranking.…”

“…Although graph drawing is the main (and most developed) application area, MLVO can also be interesting in other, very diverse, application fields like scheduling and multiple ranking. Herein, we focus on the graph drawing issue, and refer to [4] for short descriptions of the latter.…”

“…In Section 4 we show how to adopt commonly known MLCM paradigms in order to obtain simple heuristics to solve MLVO in practice and for large graphs. We conclude this section with presenting a sophisticated exact approach based on semi-definite programming (SDP), which dominates any exact approaches based on integer linear programs (ILPs) or quadratic programming (QP)-we refer the reader to the more mathematically oriented companion paper [4] for details on this approach 1 . Herein, we are mainly interested in the SDP's application to the graph drawing scenario.…”

“…Herein, we only very briefly outline an exact approach for MLVO based on semidefinite programming (SDP). We refer to [4] for the companion paper that deals almost exclusively with ILP, QP, and SDP aspects of the problem. (For the reviewer's convenience, Appendix A.2 gives a more detailed overview on our SDP.)…”