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Hardness of Packing, Covering, and Partitioning Simple Polygons with Unit Squares


Core Concepts
Packing axis-aligned unit squares into a simple polygon is NP-hard, even when the polygon is orthogonal and orthogonally convex with half-integer coordinates. The problems of finding minimum covers and partitions of simple polygons using small polygons are also NP-hard.
Abstract
The paper presents several hardness results related to packing, covering, and partitioning simple polygons with unit squares: Packing: It is NP-hard to pack axis-aligned 2×2 unit squares into a simple grid polygon, even when the polygon is orthogonal and orthogonally convex with half-integer coordinates. The authors develop a new reduction technique from Planar-3SAT that allows them to construct a non-planar geometric realization of the formula, where binary values are represented by configurations of horizontal rows and vertical columns of squares that can intersect in the interior of the polygon. Covering and Partitioning: It is NP-hard to find the minimum number of small polygons (contained in a 2×2 unit square) whose union is a given simple grid polygon (Small-Cover). It is NP-hard to find the minimum number of pairwise interior-disjoint small polygons whose union is a given simple grid polygon (Small-Partitioning). The authors show that if a polygon has an optimal covering using small polygons, then it can be partitioned into the same number of small polygons. Technical Contributions: The authors introduce the concept of "reference centers" to constrain the local structure of the packing. They develop new gadgets, such as PUSH and OR gadgets, to propagate binary values and dependencies between variables in the reduction. The construction for orthogonally convex polygons is significantly more involved, requiring a new problem called Clover-3SAT and a more complex verification of the variable components.
Stats
"Packing axis-aligned unit squares into a polygon with holes is NP-hard." "Baur and Fekete conjectured in 2001 that for any fixed integer s > 1, there is a polynomial-time algorithm to pack a maximum number of s × s squares in a simple grid polygon."
Quotes
"We show that packing axis-aligned unit squares into a simple polygon P is NP-hard, even when P is an orthogonal and orthogonally convex polygon with half-integer coordinates." "We prove that it is NP-hard to find a minimum number of small polygons whose union is P (covering) and to find a minimum number of pairwise interior-disjoint small polygons whose union is P (partitioning), when P is an orthogonal simple polygon with half-integer coordinates."

Deeper Inquiries

How can the techniques developed in this paper be extended to other packing, covering, and partitioning problems involving more complex shapes or higher-dimensional spaces

The techniques developed in the paper can be extended to other packing, covering, and partitioning problems involving more complex shapes or higher-dimensional spaces by adapting the construction of gadgets and components to suit the specific requirements of the problem. For instance, in packing problems with more complex shapes such as circles or irregular polygons, the gadgets and components can be designed to accommodate the unique characteristics of these shapes. This may involve creating specialized gadgets that interact with the boundaries of the shapes in a way that ensures a perfect packing, covering, or partitioning. In higher-dimensional spaces, the concepts of reference centers, variable components, and dependency gadgets can be generalized to work with higher-dimensional shapes. For example, in three-dimensional packing problems, the reference centers could be extended to reference volumes, and the gadgets could be designed to operate in three dimensions. The key is to adapt the construction principles used in the paper to the specific geometric properties and dimensions of the shapes involved in the problem. Overall, the techniques developed in the paper provide a framework for tackling packing, covering, and partitioning problems in various geometric settings, and with some modifications and extensions, they can be applied to a wide range of problems involving more complex shapes and higher-dimensional spaces.

Are there any special classes of simple polygons for which packing, covering, or partitioning with unit squares can be solved efficiently

There may be special classes of simple polygons for which packing, covering, or partitioning with unit squares can be solved efficiently. One potential class is that of regular polygons, such as squares, rectangles, or equilateral triangles. For regular polygons, the symmetry and uniformity of the shape may allow for more straightforward algorithms to determine optimal packings, coverings, or partitions with unit squares. The regularity of these polygons can lead to efficient algorithms that exploit the geometric properties of the shapes to find optimal solutions. Additionally, convex polygons could also be a special class for which efficient solutions are possible. Convex polygons have well-defined properties that can simplify the packing, covering, or partitioning process. Algorithms that leverage the convexity of the polygons to optimize the placement of unit squares may lead to efficient solutions for these specific cases. Moreover, polygons with certain symmetries or specific configurations, such as L-shapes or T-shapes, may also lend themselves to efficient solutions due to the constraints imposed by their geometric properties. By exploiting the inherent characteristics of these special classes of polygons, it may be possible to develop algorithms that can solve packing, covering, or partitioning problems with unit squares efficiently.

What are the implications of these hardness results for practical applications in areas like manufacturing, shipping, or VLSI design

The hardness results presented in the paper have significant implications for practical applications in areas like manufacturing, shipping, and VLSI design. In manufacturing, where efficient packing of items into containers is crucial for optimizing space and reducing costs, the NP-hardness of packing problems with unit squares highlights the complexity of finding optimal solutions. Manufacturers may need to rely on heuristic algorithms or approximation methods to address packing challenges efficiently. In shipping and logistics, where the efficient packing of goods into containers is essential for maximizing cargo capacity and minimizing transportation costs, the NP-hardness of packing, covering, and partitioning problems with unit squares underscores the challenges involved in optimizing container utilization. Shipping companies may need to invest in advanced optimization software to tackle these complex problems effectively. In VLSI design, where the placement of components on a chip is critical for ensuring proper functionality and performance, the NP-hardness of partitioning problems with unit squares indicates the computational complexity of achieving optimal chip layouts. Designers may need to employ sophisticated algorithms and tools to address partitioning challenges and meet design constraints efficiently. Overall, the hardness results emphasize the need for advanced computational methods and tools to tackle packing, covering, and partitioning problems in practical applications, highlighting the importance of algorithmic efficiency and optimization in various industries.
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