A heuristic for bottleneck crossing minimization and its performance on general crossing minimization: Hypothesis and experimental study

Extensive research over the last 20 or more years has been devoted to the problem of minimizing the total number of crossings in layered directed acyclic graphs (dags). Algorithms for this problem are used for graph drawing, to implement one of the stages in the multistage approach proposed by Sugiyama et al. [1981]. In some applications, such as minimizing the deleterious effects of crosstalk in VLSI circuits, it may be more appropriate to minimize the maximum number of crossings over all the edges. We refer to this as the bottleneck crossing problem. This article proposes a new heuristic, maximum crossings edge (MCE), designed specifically for the bottleneck problem. It is no surprise that MCE universally outperforms other heuristics with respect to bottleneck crossings. What is surprising, and the focus of this article, is that, in many settings, the MCE heuristic excels at minimizing the total number of crossings. Experiments on sparse graphs support the hypothesis that MCE gives better results (vis a vis barycenter) when the maximum degree of the dag is large. In contrast to barycenter, the number of crossings yielded by MCE is further reduced as runtime is increased. Even better results are obtained when the two heuristics are combined and/or barycenter is followed by the sifting heuristic reported in Matuszewski et al. [1999].