Let $P$ be an orthogonal polygon of $n$ vertices, without holes. The Orthogonal Polygon Covering with Squares (OPCS) problem takes as input such an orthogonal polygon $P$ with integral vertex coordinates, and asks to find the minimum number of axis-parallel squares whose union is $P$ itself. [Aupperle et. al, 1988] provide an $\mathcal O(N^{1.5})$-time algorithm for OPCS, where $N$ is the number of integral lattice points lying in $P$. In their paper, designing algorithms for OPCS with a running time polynomial in $n$, was stated as an open question; $N$ can be arbitrarily larger than $n$. Output sensitive algorithms were known due to [Bar-Yehuda and Ben-Chanoch, 1994], but these fail to address the open question, as the output can be arbitrarily larger than $n$. We address this open question by designing a polynomial-time exact algorithm for OPCS with a worst-case running time of $\mathcal O(n^{10})$. We also consider the following structural parameterized version of the problem. Let a knob be a polygon edge whose both endpoints are convex polygon vertices. Given an input orthogonal polygon without holes that has $n$ vertices and at most $k$ knobs, we design an algorithm for OPCS with a worst-case running time $\mathcal O(n^2 + k^{10} \cdot n)$. This algorithm is more efficient than the former, whenever $k = o(n^{9/10})$. The problem of Orthogonal Polygon with Holes Covering with Squares (OPCSH) is also studied by [Aupperle et. al, 1988], where the input polygon could have holes. They claim a proof that OPCSH is NP-complete even when the input is the $N$ lattice points inside the polygon. We think there is an error in their proof, where an incorrect reduction from Planar 3-CNF is shown. We provide a correct reduction with a novel construction of one of the gadgets, and show how this leads to a correct proof of NP-completeness of OPCSH.

翻译：令 $P$ 为一个不含洞的正交多边形，具有 $n$ 个顶点。正交多边形正方形覆盖（OPCS）问题以这样一个具有整数顶点坐标的正交多边形 $P$ 作为输入，要求找到最少数量的轴平行正方形，使得这些正方形的并集恰好为 $P$。[Aupperle 等人，1988] 提出了一个时间复杂度为 $\mathcal O(N^{1.5})$ 的算法来解决 OPCS 问题，其中 $N$ 是位于 $P$ 内部的整数格点数量。在他们的论文中，设计一个运行时间关于 $n$ 呈多项式的 OPCS 算法被列为开放问题；因为 $N$ 可能任意大于 $n$。[Bar-Yehuda 和 Ben-Chanoch，1994] 提出了输出敏感的算法，但这些算法未能解决该开放问题，因为输出大小也可能任意大于 $n$。我们通过设计一个最坏情况运行时间为 $\mathcal O(n^{10})$ 的多项式时间精确算法，解决了这一开放问题。我们还考虑了该问题的以下结构参数化版本。定义“旋钮”为两个端点均为凸多边形顶点的多边形边。给定一个不含洞的输入正交多边形，其具有 $n$ 个顶点和至多 $k$ 个旋钮，我们设计了一个最坏情况运行时间为 $\mathcal O(n^2 + k^{10} \cdot n)$ 的 OPCS 算法。当 $k = o(n^{9/10})$ 时，该算法比前者更高效。[Aupperle 等人，1988] 也研究了带洞正交多边形正方形覆盖（OPCSH）问题，其中输入多边形可能包含洞。他们声称证明了即使输入是多边形内部的 $N$ 个格点，OPCSH 也是 NP 完全的。我们认为他们的证明存在错误，其中展示了一个从平面 3-CNF 问题的不正确归约。我们通过新颖地构造其中一个“构件”，提供了一个正确的归约，并展示了这如何导向 OPCSH 的 NP 完全性的正确证明。