Tight Approximation Algorithms for Two Dimensional Guillotine Strip Packing
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by
Arindam Khan, Aditya Lonkar, Arnab Maiti, Amatya Sharma, Andreas Wiese
2022
Abstract
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.
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