TY - JOUR

T1 - Stronger hardness results on the computational complexity of picross 3D

AU - Kimura, Kei

N1 - Publisher Copyright:
Copyright © 2020 The Institute of Electronics, Information and Communication Engineers.

PY - 2020

Y1 - 2020

N2 - Picross 3D is a popular single-player puzzle video game for the Nintendo DS. It presents a rectangular parallelepiped (i.e., rectangular box) made of unit cubes, some of which must be removed to construct an object in three dimensions. Each row or column has at most one integer on it, and the integer indicates how many cubes in the corresponding 1D slice remain when the object is complete. Kusano et al. showed that Picross 3D is NP-complete and Kimura et al. showed that the counting version, the another solution problem, and the fewest clues problem of Picross 3D are #P-complete, NP-complete, and ΣP 2-complete, respectively, where those results are shown for the restricted input that the rectangular parallelepiped is of height four. On the other hand, Igarashi showed that Picross 3D is NPcomplete even if the height of the input rectangular parallelepiped is one. Extending the result by Igarashi, we in this paper show that the counting version, the another solution problem, and the fewest clues problem of Picross 3D are #P-complete, NP-complete, and ΣP 2 complete, respectively, even if the height of the input rectangular parallelepiped is one. Since the height of the rectangular parallelepiped of any instance of Picross 3D is at least one, our hardness results are best in terms of height.

AB - Picross 3D is a popular single-player puzzle video game for the Nintendo DS. It presents a rectangular parallelepiped (i.e., rectangular box) made of unit cubes, some of which must be removed to construct an object in three dimensions. Each row or column has at most one integer on it, and the integer indicates how many cubes in the corresponding 1D slice remain when the object is complete. Kusano et al. showed that Picross 3D is NP-complete and Kimura et al. showed that the counting version, the another solution problem, and the fewest clues problem of Picross 3D are #P-complete, NP-complete, and ΣP 2-complete, respectively, where those results are shown for the restricted input that the rectangular parallelepiped is of height four. On the other hand, Igarashi showed that Picross 3D is NPcomplete even if the height of the input rectangular parallelepiped is one. Extending the result by Igarashi, we in this paper show that the counting version, the another solution problem, and the fewest clues problem of Picross 3D are #P-complete, NP-complete, and ΣP 2 complete, respectively, even if the height of the input rectangular parallelepiped is one. Since the height of the rectangular parallelepiped of any instance of Picross 3D is at least one, our hardness results are best in terms of height.

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U2 - 10.1587/transfun.2019EAP1101

DO - 10.1587/transfun.2019EAP1101

M3 - Article

AN - SCOPUS:85082756130

VL - E103A

SP - 668

EP - 676

JO - IEICE Transactions on Fundamentals of Electronics, Communications and Computer Sciences

JF - IEICE Transactions on Fundamentals of Electronics, Communications and Computer Sciences

SN - 0916-8508

IS - 4

ER -