The authors address the geometric knapsack problem, where the items and bins are geometric objects such as squares, rectangles, and spheres. They present a framework that provides PTAS for various versions of the problem, including the hypersphere multiple knapsack problem and the multiple knapsack problem for a wide range of convex fat objects.
The key ideas are:
Structural Lemmas for Circle Packing: The authors show the existence of a super-optimal and well-structured packing of circles, which allows them to obtain a PTAS from a resource augmentation scheme.
Resource Augmentation Scheme for Circle Knapsack: The authors present a resource augmentation scheme for the circle knapsack problem, where they pack the circles in an augmented knapsack while preserving the optimality of the solution.
Generalization to Fat Objects: The authors extend their framework to handle a wider range of convex fat objects, such as ellipsoids, rhombi, and hyperspheres under the Lp-norm. They show that their resource augmentation scheme can be adapted to these objects, still yielding a PTAS.
Handling Rotations: The authors further improve their resource augmentation schemes to allow rotation of the fat objects by any angle, which is an important extension as most previous results were limited to translations.
Extensions to Other Packing Problems: The authors show that their framework can be applied to other packing problems, such as the minimum-size bin packing problem, the multiple strip packing problem, and the cutting stock problem.
The authors' framework provides a unified approach to obtain PTAS for a variety of geometric packing problems, handling a broad class of shapes and constraints.
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by Víto... klokken arxiv.org 05-02-2024
https://arxiv.org/pdf/2405.00246.pdfDypere Spørsmål