2-正则图上的开槽在线单边交叉最小化问题
The Slotted Online One-Sided Crossing Minimization Problem on 2-Regular Graphs
摘要：
在图形绘制领域，单边交叉最小化问题（OSCM）被定义在一个二分图上，其中两个顶点集彼此平行对齐，所有边都被绘制为直线。该任务是找到一个置换的节点集，使所有的边缘交叉点，称为交叉点的总数最小化。通常，一个集合的节点的度由某个常数k限制，因此该问题缩写为OSCM-k。在这项工作中，我们研究了这个问题的一个在线变体，其中一个节点集已经给出。另一个节点集和关联边被迭代地显示出来，每个节点都必须插入到占位符中，我们称之为插槽。目标再次是最小化最终图中的交叉点数量。以在线方式最小化交叉与动态图形绘制的更多经验领域有关。注意，开槽OSCM问题使得在线算法更难解决实例，但在离线情况下，它相当于没有插槽的版本。我们表明，在线开槽OSCM-k是没有竞争力的任何k大于或等于2，随后限制的2-正则图，我们显示了一个下界的4/3和上界5的竞争比的图类。
英文摘要：
In the area of graph drawing, the One-Sided Crossing Minimization Problem (OSCM) is defined on a bipartite graph with both vertex sets aligned parallel to each other and all edges being drawn as straight lines. The task is to find a permutation of one of the node sets such that the total number of all edge-edge intersections, called crossings, is minimized. Usually, the degree of the nodes of one set is limited by some constant k, with the problem then abbreviated to OSCM-k. In this work, we study an online variant of this problem, in which one of the node sets is already given. The other node set and the incident edges are revealed iteratively and each node has to be inserted into placeholders, which we call slots. The goal is again to minimize the number of crossings in the final graph. Minimizing crossings in an online way is related to the more empirical field of dynamic graph drawing. Note the slotted OSCM problem makes instances harder to solve for an online algorithm but in the offline case it is equivalent to the version without slots. We show that the online slotted OSCM-k is not competitive for any k greater or equal 2 and subsequently limit the graph class to that of 2-regular graphs, for which we show a lower bound of 4/3 and an upper bound of 5 on the competitive ratio.
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