Graph Crossing Number is a fundamental problem with various applications. In this problem, the goal is to draw an input graph $G$ in the plane so as to minimize the number of crossings between the images of its edges. Despite extensive work, non-trivial approximation algorithms are only known for bounded-degree graphs. Even for this special case, the best current algorithm achieves a $\tilde O(\sqrt n)$-approximation, while the best current negative result is APX-hardness. All current approximation algorithms for the problem build on the same paradigm: compute a set $E'$ of edges (called a \emph{planarizing set}) such that $G\setminus E'$ is planar; compute a planar drawing of $G\setminus E'$; then add the drawings of the edges of $E'$ to the resulting drawing. Unfortunately, there are examples of graphs, in which any implementation of this method must incur $\Omega (\text{OPT}^2)$ crossings, where $\text{OPT}$ is the value of the optimal solution. This barrier seems to doom the only known approach to designing approximation algorithms for the problem, and to prevent it from yielding a better than $O(\sqrt n)$-approximation. In this paper we propose a new paradigm that allows us to overcome this barrier. We show an algorithm that, given a bounded-degree graph $G$ and a planarizing set $E'$ of its edges, computes another set $E''$ with $E'\subseteq E''$, such that $|E''|$ is relatively small, and there exists a near-optimal drawing of $G$ in which only edges of $E''$ participate in crossings. This allows us to reduce the Crossing Number problem to \emph{Crossing Number with Rotation System} -- a variant in which the ordering of the edges incident to every vertex is fixed as part of input. We show a randomized algorithm for this new problem, that allows us to obtain an $O(n^{1/2-\epsilon})$-approximation for Crossing Number on bounded-degree graphs, for some constant $\epsilon>0$.

中文翻译：

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更好地逼近图交叉数

图交叉数是各种应用程序的基本问题。在这个问题中，目标是在平面上绘制一个输入图 $G$，以尽量减少其边缘图像之间的交叉次数。尽管做了大量工作，但非平凡的近似算法仅适用于有界度图。即使对于这种特殊情况，当前最好的算法也达到了 $\tilde O(\sqrt n)$-近似，而当前最好的负结果是 APX-hardness。该问题的所有当前近似算法都建立在相同的范式上：计算边的集合 $E'$（称为 \emph{planarizing set}），使得 $G\setminus E'$ 是平面的；计算 $G\setminus E'$ 的平面图；然后将 $E'$ 边缘的图形添加到生成的图形中。不幸的是，有图表的例子，其中此方法的任何实现都必须产生 $\Omega (\text{OPT}^2)$ 交叉，其中 $\text{OPT}$ 是最优解的值。这个障碍似乎注定了为该问题设计近似算法的唯一已知方法，并阻止它产生比 $O(\sqrt n)$-近似更好的方法。在本文中，我们提出了一种新范式，可以让我们克服这一障碍。我们展示了一个算法，给定一个有界度图 $G$ 和一个平面化集 $E'$ 的边，用 $E'\subseteq E''$ 计算另一个集合 $E''$，使得 $| E''|$ 相对较小，存在一个接近最优的$G$ 绘图，其中只有$E''$ 的边参与交叉。这使我们能够将交叉数问题简化为 \emph{Crossing Number with Rotation System}——一种变体，其中与每个顶点相关的边的顺序作为输入的一部分是固定的。我们展示了一个针对这个新问题的随机算法，它允许我们获得一个 $O(n^{1/2-\epsilon})$-近似于有界度图上的交叉数，对于一些常数 $\epsilon>0 $.