A PTAS for Packing Hypercubes into a Knapsack

Jansen, K and Khan, A and Lira, M and Sreenivas, KVN (2022) A PTAS for Packing Hypercubes into a Knapsack. In: 49th EATCS International Conference on Automata, Languages, and Programming, ICALP 2022, 4 - 8 July 2022, Paris.

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Abstract

We study the d-dimensional hypercube knapsack problem (d-D Hc-Knapsack) where we are given a set of d-dimensional hypercubes with associated profits, and a knapsack which is a unit d-dimensional hypercube. The goal is to find an axis-aligned non-overlapping packing of a subset of hypercubes such that the profit of the packed hypercubes is maximized. For this problem, Harren (ICALP'06) gave an algorithm with an approximation ratio of (1 + 1/2d + ε). For d = 2, Jansen and Solis-Oba (IPCO'08) showed that the problem admits a polynomial-time approximation scheme (PTAS); Heydrich and Wiese (SODA'17) further improved the running time and gave an efficient polynomial-time approximation scheme (EPTAS). Both the results use structural properties of 2-D packing, which do not generalize to higher dimensions. For d > 2, it remains open to obtain a PTAS, and in fact, there has been no improvement since Harren's result. We settle the problem by providing a PTAS. Our main technical contribution is a structural lemma which shows that any packing of hypercubes can be converted into another structured packing such that a high profitable subset of hypercubes is packed into a constant number of special hypercuboids, called V-Boxes and N-Boxes. As a side result, we give an almost optimal algorithm for a variant of the strip packing problem in higher dimensions. This might have applications for other multidimensional geometric packing problems.

Item Type:	Conference Paper
Publication:	Leibniz International Proceedings in Informatics, LIPIcs
Publisher:	Schloss Dagstuhl- Leibniz-Zentrum fur Informatik GmbH, Dagstuhl Publishing
Additional Information:	The copyright for this article belongs to the Schloss Dagstuhl- Leibniz-Zentrum fur Informatik GmbH, Dagstuhl Publishing.
Keywords:	Approximation algorithms; Combinatorial optimization; Geometry; Polynomial approximation, Approximation ratios; Cube packing; Geometric packing; Higher dimensions; Hyper-cubes; Knapsack problems; Knapsacks; Multidimensional knapsack; Polynomial time approximation schemes; Strip packing, Profitability
Department/Centre:	Division of Electrical Sciences > Computer Science & Automation
Date Deposited:	20 Jul 2022 12:03
Last Modified:	20 Jul 2022 12:03
URI:	https://eprints.iisc.ac.in/id/eprint/74942

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