有时需要两个无理数卫兵
Sometimes Two Irrational Guards are Needed
AI总结：
研究探讨了在艺术 gallery 问题中，使用两个卫兵时可能需要无理数坐标的情况，填补了关于无理数卫兵必要性的理论空白。
摘要：
在艺术 gallery 问题中，给定一个闭合多边形 $P$，其坐标为有理数，以及一个整数 $k$。我们被问是否可以找到一个大小为 $k$ 的守卫集合 $G$，使得多边形 $P$ 中的任意点 $p$ 都能被 $G$ 中的某点看到。我们说两点 $p$、$q$ 能互相看到，如果线段 $pq$ 完全位于 $P$ 内部。Abrahamsen、Adamaszek 和 Miltzow 证明存在一个多边形可以用三个守卫来守护，但若守卫必须有有理数坐标，则需要四个守卫。换句话说，最优解大小为三时可能需要无理数。我们证明最优解大小为二时也可能需要无理数。注意，任何可以用一个守卫守护的多边形，其最优守卫位置都有有理数坐标。因此，我们的工作填补了关于何时无理数守卫可能发生的理论空白。
英文摘要：
In the art gallery problem, we are given a closed polygon $P$, with rational coordinates and an integer $k$. We are asked whether it is possible to find a set (of guards) $G$ of size $k$ such that any point $p\in P$ is seen by a point in $G$. We say two points $p$, $q$ see each other if the line segment $pq$ is contained inside $P$. It was shown by Abrahamsen, Adamaszek, and Miltzow that there is a polygon that can be guarded with three guards, but requires four guards if the guards are required to have rational coordinates. In other words, an optimal solution of size three might need to be irrational. We show that an optimal solution of size two might need to be irrational. Note that it is well-known that any polygon that can be guarded with one guard has an optimal guard placement with rational coordinates. Hence, our work closes the gap on when irrational guards are possible to occur.
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