An effective crossing minimisation heuristic based on star insertion

Abstract

We present a new heuristic method for minimising crossings in a graph. The method is based upon repeatedly solving the so-called star insertion problem in the setting where the combinatorial embedding is fixed, and has several desirable characteristics for practical use. We introduce the method, discuss some aspects of algorithm design for our implementation, and provide some experimental results. The results indicate that our method compares well to existing methods, and also that it is suitable for dense instances.

Star-Struck by Fixed Embeddings: Modern Crossing Number Heuristics

“…Our in-depth experimental evaluation not only corroborates the results of previous papers [10,17] but also provides new insights into the performance of star insertion in crossing minimization heuristics. We presented the novel heuristic mim, which proceeds similarly to the planarization method but inserts most edges by reinserting one of their endpoints as a star.…”

“…We may hence call any such undesired crossings non-simple. Surprisingly, earlier implementations of the planarization method did not consider the emergence and removal of any non-simple crossings [10] while the implementation of the star reinsertion method by Clancy et al only considers β-but not α-crossings [17]. However, we show in Figure 1 that incrementally solving the same kind of insertion problem may result in a planarization with α-or β-crossings, even when starting with a planar subgraph.…”

“…The chordless cycle method (ccm) realizes the idea of extending a vertex-induced planar subgraph to a full planarization via star insertion [13]. It corresponds to the best-performing scheme for the star insertion algorithm as examined by Clancy et al [17]: Search for a chordless cycle in G, e.g., via breadth-first search. Let G denote the subgraph of G that is already embedded and initialize it with this chordless cycle.…”

“…Herein, we evaluate the aforementioned heuristics not only on their own but also in combination with the star reinsertion method (srm) by Clancy et al [17], a postprocessing strategy based on star insertion. It starts with an already existing planarization, which may be constructed using any of the methods outlined above (or even more trivial ones, such as extracting a planarization from a circular layout of the vertices, which, however, is known to perform worse [17]). To represent that the result of an algorithm is improved via srm, we append "srm" to its abbreviation, e.g.…”

“…Several variants of the planarization method have been thoroughly evaluated, including different edge insertion algorithms and postprocessing strategies; see [10] for the latest study. In a recent paper [17], Clancy et al present an alternative heuristic-the star reinsertion method -, which differs in two key aspects from the planarization method: It (i) starts with a full planarization (instead of a planar subgraph) that is iteratively improved by reinserting elements, and (ii) the reinserted elements are stars (vertices with their incident edges) rather than individual edges. These star insertions are performed using a straight-forward but never tried algorithm from literature [13].…”