We show that packing axis-aligned unit squares into a simple polygon $P$ is NP-hard, even when $P$ is an orthogonal and orthogonally convex polygon with half-integer coordinates. It has been known since the early 80s that packing unit squares into a polygon with holes is NP-hard~[Fowler, Paterson, Tanimoto, Inf. Process. Lett., 1981], but the version without holes was conjectured to be polynomial-time solvable more than two decades ago~[Baur and Fekete, Algorithmica, 2001]. Our reduction relies on a new way of reducing from \textsc{Planar-3SAT}. Interestingly, our geometric realization of a planar formula is non-planar. Vertices become rows and edges become columns, with crossings being allowed. The planarity ensures that all endpoints of rows and columns are incident to the outer face of the resulting drawing. We can then construct a polygon following the outer face that realizes all the logic of the formula geometrically, without the need of any holes. This new reduction technique proves to be general enough to also show hardness of two natural covering and partitioning problems, even when the input polygon is simple. We say that a polygon $Q$ is \emph{small} if $Q$ is contained in a unit square. We prove that it is NP-hard to find a minimum number of small polygons whose union is $P$ (covering) and to find a minimum number of pairwise interior-disjoint small polygons whose union is $P$ (partitioning), when $P$ is an orthogonal simple polygon with half-integer coordinates. This is the first partitioning problem known to be NP-hard for polygons without holes, with the usual objective of minimizing the number of pieces.

翻译：我们证明，将轴对齐的单位正方形填充到简单多边形$P$中是NP困难的，即使当$P$是具有半整数坐标的正交且正交凸多边形时也是如此。自80年代初以来，人们已知将单位正方形填充到带孔多边形中是NP困难的[Fowler, Paterson, Tanimoto, Inf. Process. Lett., 1981]，但二十多年前，无孔版本被认为可以在多项式时间内求解[Baur and Fekete, Algorithmica, 2001]。我们的归约依赖于一种从\textsc{Planar-3SAT}归约的新方法。有趣的是，我们的平面公式几何实现是非平面的。顶点变为行，边变为列，并允许交叉。平面性确保所有行和列的端点都与所得绘制的外表面相邻。然后，我们可以沿着外表面构造一个多边形，在几何上实现公式的所有逻辑，而无需任何孔洞。这种新的归约技术被证明足够通用，还可以展示两个自然覆盖和划分问题的难度，即使输入多边形是简单的。我们称多边形$Q$是\emph{小}的，如果$Q$包含在一个单位正方形内。我们证明，当$P$是具有半整数坐标的正交简单多边形时，找到覆盖$P$（覆盖）所需的最小数量的小多边形，以及找到两两内部不交且并集为$P$（划分）的最小数量的小多边形，都是NP困难的。这是第一个以最小化碎片数量为通常目标的已知无孔多边形NP困难划分问题。