The aim in packing problems is to decide if a given set of pieces can be placed inside a given container. A packing problem is defined by the types of pieces and containers to be handled, and the motions that are allowed to move the pieces. The pieces must be placed so that in the resulting placement, they are pairwise interior-disjoint. We establish a framework which enables us to show that for many combinations of allowed pieces, containers and motions, the resulting problem is $\exists \mathbb{R}$-complete. This means that the problem is equivalent (under polynomial time reductions) to deciding whether a given system of polynomial equations and inequalities with integer coefficients has a real solution. We consider packing problems where only translations are allowed as the motions, and problems where arbitrary rigid motions are allowed, i.e., both translations and rotations. When rotations are allowed, we show that it is an $\exists \mathbb{R}$-complete problem to decide if a set of convex polygons, each of which has at most $7$ corners, can be packed into a square. Restricted to translations, we show that the following problems are $\exists \mathbb{R}$-complete: (i) pieces bounded by segments and hyperbolic curves to be packed in a square, and (ii) convex polygons to be packed in a container bounded by segments and hyperbolic curves.

翻译：布局问题的目标是判定给定的一组物件能否放置在给定容器内。布局问题由待处理的物件与容器类型以及允许移动物件的运动方式共同定义。物件必须被放置使最终布局中任意两个物件内部互不相交。我们建立了一个框架，能证明对于多种允许的物件、容器和运动组合，相应的布局问题均为$\exists\mathbb{R}$完全问题。这意味着该问题（在多项式时间归约下）等价于判定一个具有整数系数的多项式方程组与不等式组是否存在实数解。我们考察了仅允许平移运动的布局问题，以及允许任意刚体运动（即同时包含平移和旋转）的问题。当允许旋转时，我们证明判定一组最多含7个顶点的凸多边形能否装入正方形是$\exists\mathbb{R}$完全问题。在仅限平移运动的情形下，我们证明以下问题为$\exists\mathbb{R}$完全问题：(i) 由线段与双曲线弧界定的物件装入正方形容器；(ii) 凸多边形装入由线段与双曲线弧界定的容器。