TY - JOUR
T1 - Packing squares independently
AU - Wu, Wei
AU - Numaguchi, Hiroki
AU - Halman, Nir
AU - Hu, Yannan
AU - Yagiura, Mutsunori
N1 - Publisher Copyright:
© 2024 Elsevier B.V.
PY - 2025/1/12
Y1 - 2025/1/12
N2 - 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.
AB - 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.
KW - Complexity
KW - Dynamic programming
KW - Fully polynomial-time approximation scheme
KW - Strip packing
UR - http://www.scopus.com/inward/record.url?scp=85206333661&partnerID=8YFLogxK
U2 - 10.1016/j.tcs.2024.114910
DO - 10.1016/j.tcs.2024.114910
M3 - ???researchoutput.researchoutputtypes.contributiontojournal.article???
AN - SCOPUS:85206333661
SN - 0304-3975
VL - 1024
JO - Theoretical Computer Science
JF - Theoretical Computer Science
M1 - 114910
ER -