Packing squares independently
Given a set of squares and a strip with bounded width and infinite height, we consider a square strip packaging problem, which we call the square independent packing problem (SIPP), to minimize the strip height so that all the squares are packed into independent cells separated by horizontal and ver...
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| Published in: | Theoretical computer science Vol. 1024; p. 114910 |
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| Main Authors: | , , , , |
| Format: | Journal Article |
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Elsevier B.V
12.01.2025
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| ISSN: | 0304-3975 |
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| Abstract | Given a set of squares and a strip with bounded width and infinite height, we consider a square strip packaging problem, which we call the square independent packing problem (SIPP), to minimize the strip height so that all the squares are packed into independent cells separated by horizontal and vertical partitions. For the SIPP, we first investigate efficient solution representations and propose a compact representation that reduces the search space from Ω(n!) to O(2n), with n the number of given squares, while guaranteeing that there exists a solution representation that corresponds to an optimal solution. Based on the solution representation, we show that the problem is NP-hard. To solve the SIPP, we propose a dynamic programming method that can be extended to a fully polynomial-time approximation scheme (FPTAS). We also propose three mathematical programming formulations based on different solution representations and confirm their performance through computational experiments with a mathematical programming solver. Finally, we discuss several extensions that are relevant to practical applications. |
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| AbstractList | Given a set of squares and a strip with bounded width and infinite height, we consider a square strip packaging problem, which we call the square independent packing problem (SIPP), to minimize the strip height so that all the squares are packed into independent cells separated by horizontal and vertical partitions. For the SIPP, we first investigate efficient solution representations and propose a compact representation that reduces the search space from Ω(n!) to O(2n), with n the number of given squares, while guaranteeing that there exists a solution representation that corresponds to an optimal solution. Based on the solution representation, we show that the problem is NP-hard. To solve the SIPP, we propose a dynamic programming method that can be extended to a fully polynomial-time approximation scheme (FPTAS). We also propose three mathematical programming formulations based on different solution representations and confirm their performance through computational experiments with a mathematical programming solver. Finally, we discuss several extensions that are relevant to practical applications. |
| ArticleNumber | 114910 |
| Author | Wu, Wei Yagiura, Mutsunori Numaguchi, Hiroki Hu, Yannan Halman, Nir |
| Author_xml | – sequence: 1 givenname: Wei orcidid: 0000-0002-0443-3642 surname: Wu fullname: Wu, Wei email: goi@shizuoka.ac.jp organization: Graduate School of Integrated Science and Technology, Shizuoka University, 3-5-1 Johoku, Chuo-ku, Hamamatsu, 432-8561, Shizuoka, Japan – sequence: 2 givenname: Hiroki surname: Numaguchi fullname: Numaguchi, Hiroki organization: Graduate School of Informatics, Nagoya University, Furo-cho, Chikusa-ku, Nagoya, 464-8601, Aichi, Japan – sequence: 3 givenname: Nir surname: Halman fullname: Halman, Nir organization: Alexander Kofkin Faculty of Engineering, Bar-Ilan University, 5290002, Ramat Gan, Israel – sequence: 4 givenname: Yannan surname: Hu fullname: Hu, Yannan organization: Department of Applied Mathematics, Tokyo University of Science, 1-3 Kagurazaka, Shinjuku-ku, 162-8601, Tokyo, Japan – sequence: 5 givenname: Mutsunori surname: Yagiura fullname: Yagiura, Mutsunori organization: Graduate School of Informatics, Nagoya University, Furo-cho, Chikusa-ku, Nagoya, 464-8601, Aichi, Japan |
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| Cites_doi | 10.1287/moor.1090.0391 10.1007/s10107-010-0371-9 10.1007/s10107-003-0427-1 10.1080/01605682.2019.1578914 10.1007/s10288-007-0051-7 10.1137/130925153 10.1016/j.dam.2007.04.022 10.1016/j.ejor.2005.12.047 10.1007/s10479-023-05265-x 10.1023/B:JOCO.0000038915.62826.79 10.1016/j.ejor.2004.02.020 10.1016/0377-2217(90)90350-K 10.1016/S0377-2217(02)00123-6 10.1287/trsc.1030.0085 10.1007/s00453-022-00954-8 |
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| Keywords | 52C15 Fully polynomial-time approximation scheme Dynamic programming 05B40 Strip packing Complexity |
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| References | Dyckhoff (br0010) 1990; 44 Wäscher, Haußner, Schumann (br0020) 2007; 183 Ibaraki, Imahori, Kubo, Masuda, Uno, Yagiura (br0120) 2005; 39 Imahori, Yagiura, Ibaraki (br0140) 2003; 97 Lodi, Martello, Monaci (br0040) 2002; 141 Lodi, Martello, Vigo (br0070) 2004; 8 Bezerra, Leao, Oliveira, Santos (br0050) 2020; 71 Halman, Klabjan, Li, Orlin, Simchi-Levi (br0110) 2014; 28 Gawiejnowicz, Halman, Kellerer (br0170) 2023; 326 Bettinelli, Ceselli, Righini (br0060) 2008; 6 Imahori, Yagiura, Ibaraki (br0150) 2005; 167 Halman, Klabjan, Mostagir, Orlin, Simchi-Levi (br0100) 2009; 34 Garey, Johnson (br0080) 1979 Saxena, Bonami, Lee (br0090) 2010; 124 Alon, Halman (br0160) 2022; 84 Júnior, Silva, Francescatto, Rosa, Siluk (br0030) 2022; 137 Ibaraki, Imahori, Nonobe, Sobue, Uno, Yagiura (br0130) 2008; 156 Saxena (10.1016/j.tcs.2024.114910_br0090) 2010; 124 Gawiejnowicz (10.1016/j.tcs.2024.114910_br0170) 2023; 326 Ibaraki (10.1016/j.tcs.2024.114910_br0130) 2008; 156 Wäscher (10.1016/j.tcs.2024.114910_br0020) 2007; 183 Lodi (10.1016/j.tcs.2024.114910_br0070) 2004; 8 Halman (10.1016/j.tcs.2024.114910_br0110) 2014; 28 Imahori (10.1016/j.tcs.2024.114910_br0140) 2003; 97 Dyckhoff (10.1016/j.tcs.2024.114910_br0010) 1990; 44 Lodi (10.1016/j.tcs.2024.114910_br0040) 2002; 141 Bettinelli (10.1016/j.tcs.2024.114910_br0060) 2008; 6 Halman (10.1016/j.tcs.2024.114910_br0100) 2009; 34 Garey (10.1016/j.tcs.2024.114910_br0080) 1979 Bezerra (10.1016/j.tcs.2024.114910_br0050) 2020; 71 Júnior (10.1016/j.tcs.2024.114910_br0030) 2022; 137 Imahori (10.1016/j.tcs.2024.114910_br0150) 2005; 167 Alon (10.1016/j.tcs.2024.114910_br0160) 2022; 84 Ibaraki (10.1016/j.tcs.2024.114910_br0120) 2005; 39 |
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| Snippet | Given a set of squares and a strip with bounded width and infinite height, we consider a square strip packaging problem, which we call the square independent... |
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| SubjectTerms | Complexity Dynamic programming Fully polynomial-time approximation scheme Strip packing |
| Title | Packing squares independently |
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