二分（伪）圆盘图：优于3的近似比算法
Bipartizing (Pseudo-)Disk Graphs: Approximation with a Ratio Better than 3
AI总结：
针对圆盘图二分化问题二十余年未突破的3-近似瓶颈，提出首个$(3-α)$（α>0为常数）近似算法，可推广至伪圆盘图且无需几何实现。
摘要：
在圆盘图中，每个顶点对应二维实数空间$\u211d^2$中的一个圆盘，当且仅当两个对应圆盘相交时，两个顶点之间存在边。圆盘图是一类重要的几何相交图，它推广了平面图和单位圆盘图。我们研究算法图论中的一个基础优化问题：二分化（也称为奇环横贯），针对圆盘图类别。二分化的目标是从输入图中删除最少数量的顶点，使得所得图是二分图。针对圆盘图二分化问题，一个民间流传的（多项式时间）3-近似算法源自Goemans和Williamson[Combinatorica'98]针对环击中问题的经典框架。二十多年来，该结果一直是该问题已知的最佳近似结果（事实上，即使对于单位圆盘图的二分化问题也是如此）。在本文中，我们实现了对该结果的首次改进，提出了一种针对圆盘图二分化问题的$(3-α)$-近似算法，其中α为某个大于0的常数。我们的算法可直接推广到更广泛的伪圆盘图类别。此外，我们的算法具有鲁棒性，即不需要给定输入图的几何实现。
英文摘要：
In a disk graph, every vertex corresponds to a disk in $\mathbb{R}^2$ and two vertices are connected by an edge whenever the two corresponding disks intersect. Disk graphs form an important class of geometric intersection graphs, which generalizes both planar graphs and unit-disk graphs. We study a fundamental optimization problem in algorithmic graph theory, Bipartization (also known as Odd Cycle Transversal), on the class of disk graphs. The goal of Bipartization is to delete a minimum number of vertices from the input graph such that the resulting graph is bipartite. A folklore (polynomial-time) $3$-approximation algorithm for Bipartization on disk graphs follows from the classical framework of Goemans and Williamson [Combinatorica'98] for cycle-hitting problems. For over two decades, this result has remained the best known approximation for the problem (in fact, even for Bipartization on unit-disk graphs). In this paper, we achieve the first improvement upon this result, by giving a $(3-α)$-approximation algorithm for Bipartization on disk graphs, for some constant $α>0$. Our algorithm directly generalizes to the broader class of pseudo-disk graphs. Furthermore, our algorithm is robust in the sense that it does not require a geometric realization of the input graph to be given.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
