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引用次数: 0
摘要
我们介绍了 "单人象棋 "的一般化,这是一种可在国际象棋网站(chess.com)上进行的单人游戏变体。该游戏的标准版本是在一个普通的 8 × 8 棋盘上由单人下棋,只有白棋,规则如下:每步棋都必须吃掉一个棋子,任何棋子都不得吃掉超过 2 次,如果棋盘上有王,它必须是最后一个棋子。我们将这个游戏推广到有 n 个棋子的无界棋盘,每个棋子都有一定的吃子次数。我们证明了广义独子棋是 NP-完全的,即使只剩下最多两次吃子的车也是如此。即使在初始配置中每个棋子都是皇后,并且只剩下两次吃子,它也是 NP-complete。与此相反,我们证明了广义独步棋的可解实例可以在以下情况下被完全表征:a)车在一维棋盘上下棋;b)兵在二维棋盘上下棋,且还有两个吃子。受广义独步棋的启发,我们还引入了图捕捉博弈,它涉及通过沿边的吃子清除图中的标记。这个棋局包含了由骑士下的广义单人棋。我们证明,对于无向图和 DAG 来说,图捕捉游戏是 NP-完全的。
We introduce a generalization of “Solo Chess”, a single-player variant of the game that can be played on chess.com. The standard version of the game is played on a regular 8 × 8 chessboard by a single player, with only white pieces, using the following rules: every move must capture a piece, no piece may capture more than 2 times, and if there is a King on the board, it must be the final piece. The goal is to clear the board, i.e., make a sequence of captures after which only one piece is left.
We generalize this game to unbounded boards with n pieces, each of which have a given number of captures that they are permitted to make. We show that Generalized Solo Chess is NP-complete, even when it is played by only rooks that have at most two captures remaining. It also turns out to be NP-complete even when every piece is a queen with exactly two captures remaining in the initial configuration. In contrast, we show that solvable instances of Generalized Solo Chess can be completely characterized when the game is: a) played by rooks on a one-dimensional board, and b) played by pawns with two captures left on a 2D board.
Inspired by Generalized Solo Chess, we also introduce the Graph Capture Game, which involves clearing a graph of tokens via captures along edges. This game subsumes Generalized Solo Chess played by knights. We show that the Graph Capture Game is NP-complete for undirected graphs and DAGs.
期刊介绍:
Theoretical Computer Science is mathematical and abstract in spirit, but it derives its motivation from practical and everyday computation. Its aim is to understand the nature of computation and, as a consequence of this understanding, provide more efficient methodologies. All papers introducing or studying mathematical, logic and formal concepts and methods are welcome, provided that their motivation is clearly drawn from the field of computing.