Framework for $\exists \mathbb{R}$-Completeness of Two-Dimensional Packing Problems
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 placemen...
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Published in | TheoretiCS Vol. 3 |
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Main Authors | , , |
Format | Journal Article |
Language | English |
Published |
26.04.2024
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Online Access | Get full text |
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Summary: | 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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ISSN: | 2751-4838 2751-4838 |
DOI: | 10.46298/theoretics.24.11 |