The Orthogonal Polygon Covering with Squares (OPCS) problem takes as input an orthogonal polygon $P$ without holes with $n$ vertices, where vertices have integral coordinates. The aim is to find a minimum number of axis-parallel, possibly overlapping squares which lie completely inside $P$, such that their union covers the entire region inside $P$. Aupperle et. al~\cite{aupperle1988covering} provide an $\mathcal O(N^{1.5})$-time algorithm to solve OPCS for orthogonal polygons without holes, where $N$ is the number of integral lattice points lying in the interior or on the boundary of $P$. Designing algorithms for OPCS with a running time polynomial in $n$ (the number of vertices of $P$) was discussed as an open question in \cite{aupperle1988covering}, since $N$ can be exponentially larger than $n$. In this paper we design a polynomial-time exact algorithm for OPCS with a running time of $\mathcal O(n^{14})$. We also consider the following structural parameterized version of the problem. A knob in an orthogonal polygon is a polygon edge whose both endpoints are convex polygon vertices. Given an input orthogonal polygon with $n$ vertices and $k$ knobs, we design an algorithm for OPCS with running time $\mathcal O(n^2 + k^{14} \cdot n)$. In \cite{aupperle1988covering}, the Orthogonal Polygon with Holes Covering with Squares (OPCSH) problem is also studied where orthogonal polygon could have holes, and the objective is to find a minimum square covering of the input polygon. This is shown to be NP-complete. We think there is an error in the existing proof in \cite{aupperle1988covering}, where a reduction from Planar 3-CNF is shown. We fix this error in the proof with an alternate construction of one of the gadgets used in the reduction, hence completing the proof of NP-completeness of OPCSH.

翻译：正交多边形正方形覆盖（OPCS）问题的输入是一个不含洞且具有$n$个顶点的正交多边形$P$，其顶点坐标为整数。目标是找到完全位于$P$内部、可能重叠的轴平行正方形的最小数量，使得这些正方形的并集覆盖$P$内部的整个区域。Aupperle等人~\cite{aupperle1988covering}提出了一种$\mathcal O(N^{1.5})$时间算法来解决不含洞正交多边形的OPCS问题，其中$N$是位于$P$内部或边界上的整数格点数量。由于$N$可能比$n$指数级更大，在\cite{aupperle1988covering}中讨论了设计运行时间关于$n$（$P$的顶点数）为多项式的算法作为一个开放问题。本文设计了一种多项式时间的精确算法求解OPCS，其运行时间为$\mathcal O(n^{14})$。我们还考虑了以下问题的结构参数化版本。正交多边形中的“旋钮”是指两个端点均为凸多边形顶点的多边形边。给定一个具有$n$个顶点和$k$个旋钮的输入正交多边形，我们设计了一种运行时间为$\mathcal O(n^2 + k^{14} \cdot n)$的OPCS算法。在\cite{aupperle1988covering}中，还研究了带洞正交多边形正方形覆盖（OPCSH）问题，其中正交多边形可能包含洞，目标是找到输入多边形的最小正方形覆盖。该问题被证明是NP完全的。我们认为\cite{aupperle1988covering}中现有证明存在一处错误，其中展示了从平面3-CNF问题的归约。我们通过重新构造归约中使用的一个构件来修正该证明错误，从而完成了OPCSH问题NP完全性的证明。