An SDP Approach to Multi-level Crossing Minimization

Abstract

We present an approach based on semidefinite programs (SDP) to tackle the multi-level crossing minimization problem. Thereby, we are given a layered graph (i.e., the graph's vertices are assigned to multiple parallel levels) and ask for an ordering of the nodes on their levels such that, when drawing the graph with straight lines, the resulting number of crossings is minimized. Solving this step is crucial in the probably most widely used graph drawing scheme, the socalled Sugiyama framework.The problem has received a lot of attention both in the field of heuristics and exact methods. For a long time, integer linear programming (ILP) approaches were the only exact algorithms applicable at least to small graphs. Recently, SDP formulations for the special case of two levels were proposed and dominated the ILP for dense instances.In this paper, 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 sideproduct, 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 the ILP solved, which 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 paper we do so for the traditional

Exact Approaches to Multilevel Vertical Orderings

“…Yet, one naturally may try to linearize the models. In fact, the ILPs for the MLCM problem can be seen as linearized models from the originally quadratic problem, and they are known to outperform SDP approaches for sparse graphs with density ≤ 10% (Chimani et al 2012). In this case we observe that most products of two binary variables have coefficients 0 in the objective function and can be omitted; only a few products have to be linearized.…”

“…Recall that proper MLVO can be seen as a special case, and our results are directly applicable to it as well. The linear ordering core of the following SDP formulation has been discussed in the context of MLCM (Chimani et al 2012). We recapitulate it here briefly, to be able to discuss MLVO-specific considerations.…”

“…There are some obvious ways to tighten (SDP b ), analogous to Chimani et al (2012). Since any (integer) feasible matrix Z ∈ MQO is an outer product of a vector INFORMS Journal on Computing 25(4), pp.…”

“…Buchheim et al (2010b) showed that an approach based on semidefinite programs (SDPs) is beneficial over the ILP formulation for (dense) bilevel problems, exploiting the stronger bounds of SDP relaxations for quadratic ordering problems (QOPs). Recently, Chimani et al (2012) showed how to further improve on this approach and generalize it to the multilevel case.…”

A linear edge kernel for two-layer crossing minimization

“…There are other methods for exactly solving TLCM such as integer programming [18] and semidefinite programming [19]. Our kernelization is expected to broaden the class of instances practically solvable by such methods.…”