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Track A: Algorithms, Complexity and Games

**Published in:** LIPIcs, Volume 229, 49th International Colloquium on Automata, Languages, and Programming (ICALP 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.

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)

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@InProceedings{khan_et_al:LIPIcs.ICALP.2022.80, author = {Khan, Arindam and Lonkar, Aditya and Maiti, Arnab and Sharma, Amatya and Wiese, Andreas}, title = {{Tight Approximation Algorithms for Two-Dimensional Guillotine Strip Packing}}, booktitle = {49th International Colloquium on Automata, Languages, and Programming (ICALP 2022)}, pages = {80:1--80:20}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-235-8}, ISSN = {1868-8969}, year = {2022}, volume = {229}, editor = {Boja\'{n}czyk, Miko{\l}aj and Merelli, Emanuela and Woodruff, David P.}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ICALP.2022.80}, URN = {urn:nbn:de:0030-drops-164215}, doi = {10.4230/LIPIcs.ICALP.2022.80}, annote = {Keywords: Approximation Algorithms, Two-Dimensional Packing, Rectangle Packing, Guillotine Cuts, Computational Geometry} }

Document

**Published in:** LIPIcs, Volume 189, 37th International Symposium on Computational Geometry (SoCG 2021)

In two-dimensional geometric knapsack problem, we are given a set of n axis-aligned rectangular items and an axis-aligned square-shaped knapsack. Each item has integral width, integral height and an associated integral profit. The goal is to find a (non-overlapping axis-aligned) packing of a maximum profit subset of rectangles into the knapsack. A well-studied and frequently used constraint in practice is to allow only 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 that do not intersect any item of the solution. In this paper we study approximation algorithms for the geometric knapsack problem under guillotine cut constraints. We present polynomial time (1+ε)-approximation algorithms for the cases with and without allowing rotations by 90 degrees, assuming that all input numeric data are polynomially bounded in n. In comparison, the best-known approximation factor for this setting is 3+ε [Jansen-Zhang, SODA 2004], even in the cardinality case where all items have the same profit.
Our main technical contribution is a structural lemma which shows that any guillotine packing can be converted into another structured guillotine packing with almost the same profit. In this packing, each item is completely contained in one of a constant number of boxes and 𝖫-shaped regions, inside which the items are placed by a simple greedy routine. In particular, we provide a clean sufficient condition when such a packing obeys the guillotine cut constraints which might be useful for other settings where these constraints are imposed.

Arindam Khan, Arnab Maiti, Amatya Sharma, and Andreas Wiese. On Guillotine Separable Packings for the Two-Dimensional Geometric Knapsack Problem. In 37th International Symposium on Computational Geometry (SoCG 2021). Leibniz International Proceedings in Informatics (LIPIcs), Volume 189, pp. 48:1-48:17, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2021)

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@InProceedings{khan_et_al:LIPIcs.SoCG.2021.48, author = {Khan, Arindam and Maiti, Arnab and Sharma, Amatya and Wiese, Andreas}, title = {{On Guillotine Separable Packings for the Two-Dimensional Geometric Knapsack Problem}}, booktitle = {37th International Symposium on Computational Geometry (SoCG 2021)}, pages = {48:1--48:17}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-184-9}, ISSN = {1868-8969}, year = {2021}, volume = {189}, editor = {Buchin, Kevin and Colin de Verdi\`{e}re, \'{E}ric}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.SoCG.2021.48}, URN = {urn:nbn:de:0030-drops-138474}, doi = {10.4230/LIPIcs.SoCG.2021.48}, annote = {Keywords: Approximation Algorithms, Multidimensional Knapsack, Guillotine Cuts, Geometric Packing, Rectangle Packing} }

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