### 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 k^{2}+3k+2/k^{2} 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 language | English |
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Title of host publication | Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics) |

Pages | 176-184 |

Number of pages | 9 |

Volume | 6681 LNCS |

DOIs | |

Publication status | Published - 2011 |

Event | 5th 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 2011 → May 31 2011 |

### Publication series

Name | Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics) |
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Volume | 6681 LNCS |

ISSN (Print) | 03029743 |

ISSN (Electronic) | 16113349 |

### Other

Other | 5th International Frontiers in Algorithmics Workshop and the 7th International Conference on Algorithmic Aspects in Information and Management, FAW-AAIM 2011 |
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Country | China |

City | Jinhua |

Period | 5/28/11 → 5/31/11 |

### Fingerprint

### ASJC Scopus subject areas

- Computer Science(all)
- Theoretical Computer Science

### Cite this

*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.

Research output: Chapter in Book/Report/Conference proceeding › Conference contribution

*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

}

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 -