2D knapsack: Packing squares

Min Chen, G. Dósa, Xin Han, Chenyang Zhou, Attila Benko

Research output: Chapter in Book/Report/Conference proceedingConference contribution

Abstract

In this paper, we study a two-dimensional knapsack problem: packing squares as many as possible into a unit square. Our results are the following: (i) first, we propose an algorithm called IHS(Increasing Height Shelf), and prove that the packing is optimal if there are at most 5 squares packed in an optimal packing, and this upper bound 5 is sharp; (ii) secondly, if all the items have size(side length) at most 1/k, where k ≥ 1 is a constant number, we propose a simple algorithm with an approximation ratio k2+3k+2/k2 in time O(n log n). (iii) finally, we give a PTAS for the general case, and our algorithm is much simpler than the previous approach[16].

Original languageEnglish
Title of host publicationLecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics)
Pages176-184
Number of pages9
Volume6681 LNCS
DOIs
Publication statusPublished - 2011
Event5th International Frontiers in Algorithmics Workshop and the 7th International Conference on Algorithmic Aspects in Information and Management, FAW-AAIM 2011 - Jinhua, China
Duration: May 28 2011May 31 2011

Publication series

NameLecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics)
Volume6681 LNCS
ISSN (Print)03029743
ISSN (Electronic)16113349

Other

Other5th International Frontiers in Algorithmics Workshop and the 7th International Conference on Algorithmic Aspects in Information and Management, FAW-AAIM 2011
CountryChina
CityJinhua
Period5/28/115/31/11

Fingerprint

Knapsack
Packing
Knapsack Problem
Upper bound
Unit
Approximation

ASJC Scopus subject areas

  • Computer Science(all)
  • Theoretical Computer Science

Cite this

Chen, M., Dósa, G., Han, X., Zhou, C., & Benko, A. (2011). 2D knapsack: Packing squares. In Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics) (Vol. 6681 LNCS, pp. 176-184). (Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics); Vol. 6681 LNCS). https://doi.org/10.1007/978-3-642-21204-8_21

2D knapsack : Packing squares. / Chen, Min; Dósa, G.; Han, Xin; Zhou, Chenyang; Benko, Attila.

Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics). Vol. 6681 LNCS 2011. p. 176-184 (Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics); Vol. 6681 LNCS).

Research output: Chapter in Book/Report/Conference proceedingConference contribution

Chen, M, Dósa, G, Han, X, Zhou, C & Benko, A 2011, 2D knapsack: Packing squares. in Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics). vol. 6681 LNCS, Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics), vol. 6681 LNCS, pp. 176-184, 5th International Frontiers in Algorithmics Workshop and the 7th International Conference on Algorithmic Aspects in Information and Management, FAW-AAIM 2011, Jinhua, China, 5/28/11. https://doi.org/10.1007/978-3-642-21204-8_21
Chen M, Dósa G, Han X, Zhou C, Benko A. 2D knapsack: Packing squares. In Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics). Vol. 6681 LNCS. 2011. p. 176-184. (Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics)). https://doi.org/10.1007/978-3-642-21204-8_21
Chen, Min ; Dósa, G. ; Han, Xin ; Zhou, Chenyang ; Benko, Attila. / 2D knapsack : Packing squares. Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics). Vol. 6681 LNCS 2011. pp. 176-184 (Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics)).
@inproceedings{eb5494e5918b45d8a11b49f38728606e,
title = "2D knapsack: Packing squares",
abstract = "In this paper, we study a two-dimensional knapsack problem: packing squares as many as possible into a unit square. Our results are the following: (i) first, we propose an algorithm called IHS(Increasing Height Shelf), and prove that the packing is optimal if there are at most 5 squares packed in an optimal packing, and this upper bound 5 is sharp; (ii) secondly, if all the items have size(side length) at most 1/k, where k ≥ 1 is a constant number, we propose a simple algorithm with an approximation ratio k2+3k+2/k2 in time O(n log n). (iii) finally, we give a PTAS for the general case, and our algorithm is much simpler than the previous approach[16].",
author = "Min Chen and G. D{\'o}sa and Xin Han and Chenyang Zhou and Attila Benko",
year = "2011",
doi = "10.1007/978-3-642-21204-8_21",
language = "English",
isbn = "9783642212031",
volume = "6681 LNCS",
series = "Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics)",
pages = "176--184",
booktitle = "Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics)",

}

TY - GEN

T1 - 2D knapsack

T2 - Packing squares

AU - Chen, Min

AU - Dósa, G.

AU - Han, Xin

AU - Zhou, Chenyang

AU - Benko, Attila

PY - 2011

Y1 - 2011

N2 - In this paper, we study a two-dimensional knapsack problem: packing squares as many as possible into a unit square. Our results are the following: (i) first, we propose an algorithm called IHS(Increasing Height Shelf), and prove that the packing is optimal if there are at most 5 squares packed in an optimal packing, and this upper bound 5 is sharp; (ii) secondly, if all the items have size(side length) at most 1/k, where k ≥ 1 is a constant number, we propose a simple algorithm with an approximation ratio k2+3k+2/k2 in time O(n log n). (iii) finally, we give a PTAS for the general case, and our algorithm is much simpler than the previous approach[16].

AB - In this paper, we study a two-dimensional knapsack problem: packing squares as many as possible into a unit square. Our results are the following: (i) first, we propose an algorithm called IHS(Increasing Height Shelf), and prove that the packing is optimal if there are at most 5 squares packed in an optimal packing, and this upper bound 5 is sharp; (ii) secondly, if all the items have size(side length) at most 1/k, where k ≥ 1 is a constant number, we propose a simple algorithm with an approximation ratio k2+3k+2/k2 in time O(n log n). (iii) finally, we give a PTAS for the general case, and our algorithm is much simpler than the previous approach[16].

UR - http://www.scopus.com/inward/record.url?scp=79957966373&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=79957966373&partnerID=8YFLogxK

U2 - 10.1007/978-3-642-21204-8_21

DO - 10.1007/978-3-642-21204-8_21

M3 - Conference contribution

AN - SCOPUS:79957966373

SN - 9783642212031

VL - 6681 LNCS

T3 - Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics)

SP - 176

EP - 184

BT - Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics)

ER -