We present a 1.8334-approximation algorithm for Vertex Cover on string graphs given with a representation, which takes polynomial time in the size of the representation; the exact approximation factor is $11/6$. Recently, the barrier of 2 was broken by Lokshtanov et al. [SoGC '24] with a 1.9999-approximation algorithm. Thus we increase by three orders of magnitude the distance of the approximation ratio to the trivial bound of 2. Our algorithm is very simple. The intricacies reside in its analysis, where we mainly establish that string graphs without odd cycles of length at most 11 are 8-colorable. Previously, Chudnovsky, Scott, and Seymour [JCTB '21] showed that string graphs without odd cycles of length at most 7 are 80-colorable, and string graphs without odd cycles of length at most 5 have bounded chromatic number.

翻译：本文提出一种针对给定表示的弦图顶点覆盖问题的1.8334近似算法，该算法在表示规模的多项式时间内运行；其精确近似因子为11/6。近期，Lokshtanov等人[SoGC '24]通过1.9999近似算法突破了2的障碍。因此，我们将近似比与平凡上界2的距离提升了三个数量级。本算法极为简洁，其复杂性主要体现在分析过程中：我们主要证明了不含长度至多为11的奇环的弦图是8-可染色的。此前，Chudnovsky、Scott与Seymour[JCTB '21]的研究表明：不含长度至多为7的奇环的弦图是80-可染色的，而不含长度至多为5的奇环的弦图其色数存在上界。