Conceitos Básicos
Many variants of two-dimensional packing problems, including those with curved pieces and containers, are ∃R-complete, indicating their fundamental computational difficulty.
Resumo
The paper establishes a framework for proving ∃R-completeness of a wide range of two-dimensional packing problems. The key insights are:
Packing problems can be reduced from the Existential Theory of the Reals (ETR), a fundamental problem in real algebraic geometry. This shows that many packing problems are as hard as deciding the satisfiability of systems of polynomial equations and inequalities over the reals.
The authors introduce an auxiliary problem called Wired-Curve-ETR[f, g], which is a graphical representation of an ETR formula. They show that this problem is ∃R-hard, and then reduce it to various packing problems.
The packing problems considered include allowing translations only, as well as allowing both translations and rotations. The pieces can be convex polygons, curved polygons (bounded by line segments and hyperbolic arcs), and simple polygons. The containers can be squares, convex polygons, and curved polygons.
The authors prove that 10 out of the 12 variants of packing problems they consider are ∃R-complete. This indicates the fundamental computational difficulty of these problems, as ∃R-hard problems are believed to be harder than NP-complete problems.
The reductions ensure that the constructed packing instances have pieces with constant complexity and coordinates that can be described using a logarithmic number of bits. This shows the problems are strongly ∃R-hard.
The authors also provide insights on the relationship between packing problems and the complexity class ∃R, as well as discuss open problems and directions for future research.
Estatísticas
The paper does not contain any specific numerical data or statistics. It focuses on establishing the computational complexity of various packing problems.
Citações
"The aim in packing problems is to decide if a given set of pieces can be placed inside a given container."
"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."
"We show that many of the above mentioned variants of packing are ∃R-complete. The complexity class ∃R will be defined below."