猎人与兔子游戏的PSPACE完全性
PSPACE-completeness of the Hunters and Rabbit game
AI总结：
研究猎人与兔子游戏中判定给定猎人数量能否捕获兔子的问题，证明其在简单二分图上为PSPACE完全，还推广至消防游戏等场景，丰富了组合游戏的计算复杂性研究。
摘要：
在猎人与兔子游戏中，每一轮猎人们向一组顶点射击，而一只看不见的兔子必须沿着一条边移动。我们证明，判定给定数量的猎人能否保证捕获兔子的问题是PSPACE完全的，即使是在简单二分图上也成立。该归约使用了逆有限自动机的交集非空性：一个公共单词必须同时将所有自动机置于接受状态。一个充当安全港的团和两类守卫迫使每一个获胜策略模拟公共转移。现有变换消除了对兔子初始位置的限制，并针对有一个猎人的有向游戏证明了其PSPACE完全性，即使是在最小出度为正的定向图上也成立。调整守卫的大小还证明了消防游戏的PSPACE完全性，在该游戏中兔子可以留在当前顶点。
英文摘要：
In the Hunters and Rabbit game, hunters shoot at a set of vertices at each round, while an invisible rabbit must move along an edge. We prove that deciding whether a given number of hunters can guarantee capture is PSPACE-complete, even on simple bipartite graphs. The reduction uses intersection nonemptiness for inverse finite automata: one common word must put all automata in accepting states simultaneously. A clique acting as a safe haven and two families of guards force every winning strategy to simulate common transitions. Existing transformations remove the restriction on the rabbit's initial positions and yield PSPACE-completeness for the directed game with one hunter, even on oriented graphs with positive minimum outdegree. Adjusting the guard sizes also proves PSPACE-completeness of the firefighting game, in which the rabbit may remain at its current vertex.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
