Tight Approximation Algorithms for Two-Dimensional Guillotine Strip Packing

Authors Arindam Khan , Aditya Lonkar, Arnab Maiti, Amatya Sharma, Andreas Wiese

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Author Details

Arindam Khan
  • Department of Computer Science and Automation, Indian Institute of Science, Bangalore, India
Aditya Lonkar
  • Department of Computer Science and Automation, Indian Institute of Science, Bangalore, India
Arnab Maiti
  • Indian Institute of Technology, Kharagpur, India
Amatya Sharma
  • Indian Institute of Technology, Kharagpur, India
Andreas Wiese
  • Technische Universität München, Germany


A part of this work was done when Arnab Maiti and Amatya Sharma were undergraduate interns at Indian Institute of Science.

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Arindam Khan, Aditya Lonkar, Arnab Maiti, Amatya Sharma, and Andreas Wiese. Tight Approximation Algorithms for Two-Dimensional Guillotine Strip Packing. In 49th International Colloquium on Automata, Languages, and Programming (ICALP 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 229, pp. 80:1-80:20, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2022)


In the Strip Packing problem (SP), we are given a vertical half-strip [0,W]×[0,∞) and a set of n axis-aligned rectangles of width at most W. The goal is to find a non-overlapping packing of all rectangles into the strip such that the height of the packing is minimized. A well-studied and frequently used practical constraint is to allow only those packings that are guillotine separable, i.e., every rectangle in the packing can be obtained by recursively applying a sequence of edge-to-edge axis-parallel cuts (guillotine cuts) that do not intersect any item of the solution. In this paper, we study approximation algorithms for the Guillotine Strip Packing problem (GSP), i.e., the Strip Packing problem where we require additionally that the packing needs to be guillotine separable. This problem generalizes the classical Bin Packing problem and also makespan minimization on identical machines, and thus it is already strongly NP-hard. Moreover, due to a reduction from the Partition problem, it is NP-hard to obtain a polynomial-time (3/2-ε)-approximation algorithm for GSP for any ε > 0 (exactly as Strip Packing). We provide a matching polynomial time (3/2+ε)-approximation algorithm for GSP. Furthermore, we present a pseudo-polynomial time (1+ε)-approximation algorithm for GSP. This is surprising as it is NP-hard to obtain a (5/4-ε)-approximation algorithm for (general) Strip Packing in pseudo-polynomial time. Thus, our results essentially settle the approximability of GSP for both the polynomial and the pseudo-polynomial settings.

Subject Classification

ACM Subject Classification
  • Theory of computation → Design and analysis of algorithms
  • Approximation Algorithms
  • Two-Dimensional Packing
  • Rectangle Packing
  • Guillotine Cuts
  • Computational Geometry


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