二维装箱问题 $\exists \mathbb{R}$-完全性的框架
Framework for $\exists \mathbb{R}$-Completeness of Two-Dimensional Packing Problems
AI总结：
本文建立了一个证明二维装箱问题 $\exists \mathbb{R}$-完全性的通用框架，并证明允许旋转时最多7个角的凸多边形装入正方形、以及仅平移时由线段和双曲曲线围成的物体装入正方形等问题均为 $\exists \mathbb{R}$-完全。
摘要：
装箱问题的目标是判断给定的一组物体是否能放入给定的容器中。一个装箱问题由所处理的物体类型、容器类型以及允许移动物体的运动方式所定义。物体必须被放置得在最终布局中两两内部不相交。我们建立了一个框架，使我们能够证明：对于许多允许的物体、容器和运动方式的组合，所得到的问题是 $\exists \mathbb{R}$-完全的。这意味着该问题（在多项式时间归约下）等价于判断一个给定的整数系数多项式方程与不等式组是否有实数解。我们考虑只允许平移作为运动方式的装箱问题，以及允许任意刚体运动（即同时允许平移和旋转）的问题。当允许旋转时，我们证明判断一组凸多边形（每个最多有 $7$ 个角）能否装入一个正方形是一个 $\exists \mathbb{R}$-完全问题。在仅限于平移的情况下，我们证明以下问题是 $\exists \mathbb{R}$-完全的：（i）由线段和双曲曲线围成的物体装入正方形的问题，以及（ii）凸多边形装入由线段和双曲曲线围成的容器的问题。
英文摘要：
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.
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