TY - GEN

T1 - Approximation Schemes for Geometric Knapsack for Packing Spheres and Fat Objects

AU - Acharya, Pritam

AU - Bhore, Sujoy

AU - Gupta, Aaryan

AU - Khan, Arindam

AU - Mondal, Bratin

AU - Wiese, Andreas

N1 - Publisher Copyright:
© Pritam Acharya, Sujoy Bhore, Aaryan Gupta, Arindam Khan, Bratin Mondal, and Andreas Wiese.

PY - 2024/7/1

Y1 - 2024/7/1

N2 - 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 Oε(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.

AB - 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 Oε(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.

KW - Approximation Algorithms

KW - Circle Packing

KW - Geometric Knapsack

KW - Polygon Packing

KW - Resource Augmentation

KW - Sphere Packing

UR - http://www.scopus.com/inward/record.url?scp=85198346848&partnerID=8YFLogxK

U2 - 10.4230/LIPIcs.ICALP.2024.8

DO - 10.4230/LIPIcs.ICALP.2024.8

M3 - Conference contribution

AN - SCOPUS:85198346848

T3 - Leibniz International Proceedings in Informatics, LIPIcs

BT - 51st International Colloquium on Automata, Languages, and Programming, ICALP 2024

A2 - Bringmann, Karl

A2 - Grohe, Martin

A2 - Puppis, Gabriele

A2 - Svensson, Ola

PB - Schloss Dagstuhl- Leibniz-Zentrum fur Informatik GmbH, Dagstuhl Publishing

T2 - 51st International Colloquium on Automata, Languages, and Programming, ICALP 2024

Y2 - 8 July 2024 through 12 July 2024

ER -