Abstract
While the problem of packing two-dimensional squares into a square, in which a set of squares is packed into a big square, has been proved to be NP-complete, the computational complexity of the d-dimensional (\( d\ge 3 \)) problems of packing hypercubes into a hypercube remains an open question (Acta Inf 41(9):595–606, 2005; Theor Comput Sci 410(44):4504–4532, 2009). In this paper, the authors show that the three-dimensional problem version of packing cubes into a cube is NP-complete in the strong sense.
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Notes
A cubic space or box has 6 faces, as shown in Fig. 4b, a. In such a figure, we call its visible faces that are parallel to the \(x\times y\) plane, \(y\times z\) plane, or \(z\times x\) plane the front face, the top face, or the left face, respectively, and call its invisible faces that are parallel to the \(x \times y\) plane, \(y \times z\) plane, or \(z\times x\) plane the back face, the bottom face, or the right face, respectively.
Region \(\varepsilon -\varepsilon '\) is the region \(\varepsilon \) minus the region \(\varepsilon '\), similar is the meaning of region \(\zeta -\zeta '\). See Fig. 3b for region \(\varepsilon \) and region \(\zeta \).
References
Bansal N, Correa JR, Kenyon C, Sviridenko M (2006) Bin packing in multiple dimensions: Inapproximability results and approximation schemes. Math Oper Res 31(1):31–49
Caprara A, Lodi A, Monaci M (2005) Fast approximation schemes for two-stage, two-dimensional bin packing. Math Oper Res 30:136–156
Chung FRK, Garey MR, Johnson DS (1982) On packing two-dimensional bins. SIAM J Algebraic Discret Methods 3:66–76
Correa JR, Kenyon C (2004) Approximation schemes for multidimensional packing. In: Proceedings of 15th ACM-SIAM symposium on discrete algorithms 179–188
Epstein L, van Stee R (2005) Online square and cube packing. Acta Inf 41(9):595–606
Garey M, Johnson D (1979) Computer and intractability—a guide to the theory of np-completeness. Freeman, New York
Harren R (2009) Approximation algorithms for orthogonal packing problems for hypercubes. Theor Comput Sci 410(44):4504–4532
Kohayakawa Y, Miyazawa FK, Raghavan P, Wakabayashi Y (2004) Multidimensional cube packing. Algorithmica 40:173–187
Leung JYT, Tam WT, Wong CS, Chin FYL (1990) Packing squares into a square. J Parallel Distrib Comput 10:271–275
Li K, Cheng KH (1989) Complexity of resource allocation and job scheduling problems in partitionable mesh connected systems. Proceedings of 1st annual IEEE symposium of parallel and distributed processing, Silver Spring, MD pp 358–365
Miyazawa F, Wakabayashi Y (2003) Cube packing. Theor Comput Sci 297:355–366
Waescher G, Haussner H, Schumann H (2007) An improved typology of cutting and packing problems. Euro J Oper Res 183(3):1109–1130
Acknowledgments
The research of D. Z. Chen was supported in part by NSF under Grants CCF-0916606 and CCF-1217906.
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Lu, Y., Chen, D.Z. & Cha, J. Packing cubes into a cube is NP-complete in the strong sense. J Comb Optim 29, 197–215 (2015). https://doi.org/10.1007/s10878-013-9701-1
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DOI: https://doi.org/10.1007/s10878-013-9701-1