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Approximation Schemes for Geometric Knapsack for Packing Spheres and Fat Objects

Authors Pritam Acharya, Sujoy Bhore , Aaryan Gupta, Arindam Khan , Bratin Mondal, Andreas Wiese

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Pritam Acharya
  • Department of Mathematics, Indian Institute of Science Education and Research Pune, India
Sujoy Bhore
  • Department of Computer Science and Engineering, Indian Institute of Technology Bombay, India
Aaryan Gupta
  • Department of Computer Science and Engineering, Indian Institute of Technology Bombay, India
Arindam Khan
  • Department of Computer Science and Automation, Indian Institute of Science Bengaluru, India
Bratin Mondal
  • Department of Computer Science and Engineering, Indian Institute of Technology Kharagpur, India
Andreas Wiese
  • Department of Mathematics, Technical University of Munich, Germany

Funding

Arindam Khan’s research is supported in part by Google India Research Award, SERB Core Research Grant (CRG/2022/001176) on "Optimization under Intractability and Uncertainty", and the Walmart Center for Tech Excellence at IISc (CSR Grant WMGT-23-0001). A part of this work was done when Aaryan Gupta, Bratin Mondal, and Pritam Acharya were undergraduate interns at Indian Institute of Science, supported by Arindam Khan’s Pratiksha Trust Young Investigator Award.

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Pritam Acharya, Sujoy Bhore, Aaryan Gupta, Arindam Khan, Bratin Mondal, and Andreas Wiese. Approximation Schemes for Geometric Knapsack for Packing Spheres and Fat Objects. In 51st International Colloquium on Automata, Languages, and Programming (ICALP 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 297, pp. 8:1-8:20, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024)
https://doi.org/10.4230/LIPIcs.ICALP.2024.8

Abstract

We study the geometric knapsack problem in which we are given a set of d-dimensional objects (each with associated profits) and the goal is to find the maximum profit subset that can be packed non-overlappingly into a given d-dimensional (unit hypercube) knapsack. Even if d = 2 and all input objects are disks, this problem is known to be NP-hard [Demaine, Fekete, Lang, 2010]. In this paper, we give polynomial time (1+ε)-approximation algorithms for the following types of input objects in any constant dimension d: - disks and hyperspheres, - a class of fat convex polygons that generalizes regular k-gons for k ≥ 5 (formally, polygons with a constant number of edges, whose lengths are in a bounded range, and in which each angle is strictly larger than π/2), - arbitrary fat convex objects that are sufficiently small compared to the knapsack. We remark that in our PTAS for disks and hyperspheres, we output the computed set of objects, but for a O_ε(1) of them we determine their coordinates only up to an exponentially small error. However, it is not clear whether there always exists a (1+ε)-approximate solution that uses only rational coordinates for the disks' centers. We leave this as an open problem which is related to well-studied geometric questions in the realm of circle packing.

Subject Classification

ACM Subject Classification
  • Theory of computation → Design and analysis of algorithms
Keywords
  • Approximation Algorithms
  • Polygon Packing
  • Circle Packing
  • Sphere Packing
  • Geometric Knapsack
  • Resource Augmentation

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