# Framework for ∃ℝ-Completeness of Two-Dimensional Packing Problems

We show that many natural two-dimensional packing problems are algorithmically equivalent to finding real roots of multivariate polynomials. A two-dimensional packing problem is defined by the type of pieces, containers, and motions that are allowed. The aim is to decide if a given set of pieces can be placed inside a given container. The pieces must be placed so that they are pairwise interior-disjoint, and only motions of the allowed type can be used to move them there. We establish a framework which enables us to show that for many combinations of allowed pieces, containers, and motions, the resulting problem is ∃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 the following combinations of allowed pieces and containers are ∃ℝ-complete: ∙ simple polygons, each of which has at most 8 corners, into a square, ∙ convex objects bounded by line segments and hyperbolic curves into a square, ∙ convex polygons into a container bounded by line segments and hyperbolic curves. Restricted to translations, we show that the following combinations are ∃ℝ-complete: ∙ objects bounded by segments and hyperbolic curves into a square, ∙ convex polygons into a container bounded by segments and hyperbolic curves.

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