AI 中文总结
研究如何用\(\ell_2\)度量将简单多边形覆盖和分割为具有单位测地半径或直径的部分，针对半径问题给出覆盖和分割的9近似算法，针对直径问题的分割版本给出15近似算法，改进了前人复杂算法。
AI 中文摘要
我们考虑使用\(\ell_2\)度量来覆盖和分割简单多边形，使其分割后的部分具有单位测地半径或单位测地直径。即使输入规模恒定，也尚无找到这些问题精确解的方法，且已知对于有洞多边形该问题是NP难的。因此，我们致力于开发多项式时间运行的简单近似算法。对于半径问题，给出了覆盖和分割的首个近似算法，近似因子为9。对于直径问题，仅在分割版本上取得积极成果，改进了Abrahamsen和Rasmussen复杂的72近似，实现了简单的15近似。
英文摘要
We consider covering and partitioning a simple polygon into pieces which either have unit geodesic radius or unit geodesic diameter, using the $\ell_2$-metric for distances. There is no known method for finding an exact solution to these problems, even when the input size is constant, and the problem is known to be NP-hard in the case of polygons with holes. With this in mind, we instead devote our attention to developing simple approximation algorithms that run in polynomial time. For the radius problem, we present the first known approximation algorithms for both covering and partitioning, achieving a factor of 9. For the diameter problem, we are only able to give a positive result for the partition version of the problem, where we improve upon a complicated 72-approximation from Abrahamsen and Rasmussen [SODA '25], achieving a simple 15-approximation.
