A Tight (3/2+ε) Approximation for Skewed Strip Packing

Waldo Gálvez (Corresponding author), Fabrizio Grandoni, Afrouz Jabal Ameli, Klaus Jansen, Arindam Khan, Malin Rau

Research output: Contribution to journalArticleAcademicpeer-review

2 Citations (Scopus)
66 Downloads (Pure)

Abstract

In the Strip Packing problem, we are given a vertical half-strip [0 , W] × [0 , + ∞) and a collection of open rectangles of width at most W. Our goal is to find an axis-aligned (non-overlapping) packing of such rectangles into the strip such that the maximum height OPT spanned by the packing is as small as possible. It is NP-hard to approximate this problem within a factor (3 / 2 - ε) for any constant ε> 0 by a simple reduction from the Partition problem, while the current best approximation factor for it is (5 / 3 + ε) . It seems plausible that Strip Packing admits a (3 / 2 + ε) -approximation. We make progress in that direction by achieving such tight approximation guarantees for a special family of instances, which we call skewed instances. As standard in the area, for a given constant parameter δ> 0 , we call large the rectangles with width at least δW and height at least δOPT , and skewed the remaining rectangles. If all the rectangles in the input are large, then one can easily compute the optimal packing in polynomial time (since the input can contain only a constant number of rectangles). We consider the complementary case where all the rectangles are skewed. This second case retains a large part of the complexity of the original problem; in particular, the skewed case is still NP-hard to approximate within a factor (3 / 2 - ε) , and we provide an (almost) tight (3 / 2 + ε) -approximation algorithm.

Original languageEnglish
Pages (from-to)3088-3109
Number of pages22
JournalAlgorithmica
Volume85
Issue number10
Early online date10 May 2023
DOIs
Publication statusPublished - Oct 2023

Keywords

  • Approximation algorithms
  • Rectangle packing
  • Strip packing

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