One-Sided Local Crossing Minimization

Drawing graphs with the minimum number of crossings is a classical problem that has been studied extensively. Many restricted versions of the problem have been considered. For example, bipartite graphs can be drawn such that the two sets in the bipartition of the vertex set are mapped to two parallel lines, and the edges are drawn as straight-line segments. In this setting, the number of crossings depends only on the ordering of the vertices on the two lines. Two natural variants of the problem have been studied. In the one-sided case, the order of the vertices on one of the two lines is given and fixed; in the two-sided case, no order is given. Both cases are important subproblems in the so-called Sugiyama framework for drawing layered graphs with few crossings, and both turned out to be NP-hard. For the one-sided case, Eades and Wormald [Algorithmica 1994] introduced the median heuristic and showed that it has an approximation ratio of 3. In recent years, researchers have focused on a local version of crossing minimization where the aim is not to minimize the total number of crossings but the maximum number of crossings per edge. Kobayashi, Okada, and Wolff [SoCG 2025] investigated the complexity of local crossing minimization parameterized by the natural parameter. They showed that the weighted one-sided problem is NP-hard and conjectured that the unweighted one-sided case remains NP-hard. In this work, we confirm their conjecture. We also prove that the median heuristic with a specific tie-breaking scheme has an approximation ratio of 3.