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.

翻译：绘制具有最少交叉数量的图是一个经典问题，已得到广泛研究。该问题的许多受限版本已被考虑。例如，二分图可以绘制为：顶点集二分划分中的两个子集被映射到两条平行线上，而边被绘制为直线段。在此设置下，交叉数量仅取决于两条线上顶点的排序。该问题的两个自然变体已被研究。在单侧情况下，两条线中一条线上顶点的顺序是给定且固定的；在双侧情况下，则未给定任何顺序。这两种情况都是在所谓的 Sugiyama 框架中绘制层状图时减少交叉的重要子问题，且两者均被证明是 NP 难的。对于单侧情况，Eades 和 Wormald [Algorithmica 1994] 引入了中位数启发式算法，并证明其近似比为 3。近年来，研究者们关注于交叉最小化的局部版本，其目标不是最小化总交叉数，而是最小化每条边的最大交叉数。Kobayashi、Okada 和 Wolff [SoCG 2025] 研究了以自然参数为参数的局部交叉最小化的复杂度。他们证明了加权单侧问题是 NP 难的，并推测未加权单侧情况仍然是 NP 难的。在本工作中，我们证实了他们的推测。我们还证明了采用特定平局决胜方案的中位数启发式算法具有近似比 3。