Light Up variants are NP-complete

抄録

The Light Up puzzle is one of the Nikoli's puzzles, and its NP-completeness has already been shown [8]. In this paper, we introduce a new variant of Light Up that naturally arises from its connection with the n-Queens problem, and discuss the problem of deciding whether it has a solution for this Light Up variant under various restrictions. This variant extends the directions of light from four (up, down, left, right) to eight, including diagonal directions, and this extension corresponds to the chess queen's movement in the n-Queens problem [2]. In the original Light Up puzzle, the numbers 0, 1, 2, 3, and 4 represent the number of light bulb placements. However, when considering the extension of the light irradiation direction, the number of light bulb placements is limited to 0, 1, and 2. In this paper, we analyze the computational complexity of the Light Up variant that limits the number of light bulb placements to only one type, and prove its NP-completeness using a reduction from Circuit-SAT, specifically for the cases where the number of light bulb placements is restricted to 0 or 1. These results also show that all problems are NP-complete except for the case where the number of light bulb placements is exactly 2.