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Packing cubes into a cube is NP-complete in the strong sense

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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

  1. 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.

  2. 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 \(\zeta \).

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Acknowledgments

The research of D. Z. Chen was supported in part by NSF under Grants CCF-0916606 and CCF-1217906.

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Authors and Affiliations

  1. School of Mechanical, Electronic and Control Engineering, Beijing Jiaotong University, Beijing, 100044, China

    Yiping Lu & Jianzhong Cha

  2. Department of Computer Science and Engineering, University of Notre Dame, Notre Dame, IN, 46556 , USA

    Danny Z. Chen

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  1. Yiping Lu
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  2. Danny Z. Chen
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  3. Jianzhong Cha
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Correspondence to Yiping Lu.

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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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