广义可达性游戏
Generalised Reachability Games
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中文总结 AI 辅助
研究广义可达性游戏中判定胜者的复杂性，证明其为PSPACE完全问题，目标集大小是决定复杂性的重要参数，还给出了玩家记忆需求及策略大小的界，研究了优化变体，多数情况下优化问题难处理。
中文摘要 AI 辅助
我们研究在具有多个可达性目标的图上进行的两人零和回合制游戏，即广义可达性游戏。在经典可达性游戏中，一方玩家伊芙的目标是访问给定的目标顶点集，另一方玩家亚当则阻止她。在广义可达性游戏中，单个目标集被一族目标集取代，伊芙的目标是以任意顺序访问所有目标集。我们研究了具有广义可达性目标的两人游戏中判定胜者的复杂性。研究表明，决定该问题复杂性的一个重要参数是目标集的大小。首先证明此类游戏中判定胜者是PSPACE完全问题，即使每个目标集大小至多为3时PSPACE下界也成立。相比之下，当目标集大小大于1的数量时，该问题是固定参数可处理的。此外，我们考虑了双方玩家的记忆需求，并给出了获胜策略大小的匹配上界和下界。我们还研究了这些游戏的优化变体。对于优化问题，在大多数有趣的情况下都显示出难处理性。特别是，与单元素目标集情况下广义可达性的可处理性相反，当伊芙试图最大化访问的目标集数量时，优化问题是coNP难的。在不同的优化设置中，即伊芙被要求保证能访问的目标集的最大子集时，这种情况的可处理性可以恢复。
英文摘要
We study two-player zero-sum turn-based games played on graphs with multiple reachability objectives called generalised reachability games. In classic reachability games the goal of one player, Eve, is to visit a given target set of vertices, and that of the other player, Adam, is to prevent this. In generalised reachability games, the single target set is replaced with a family of target sets and the objective of Eve is to visit all of them in any order. We study the complexity of deciding the winner in two-player games with generalised reachability objectives. Our study reveals that an important parameter that determines the complexity of this problem is the size of the target sets. We first prove that deciding the winner in such games is PSPACE-complete, and the PSPACE lower bound holds even when the size of each target set is at most three. By contrast, we show that the problem is FPT in the number of target sets of size greater than one. Moreover, we consider the memory requirements for both players and give matching upper and lower bounds on the sizes of winning strategies. We also study optimisation variants of these games. For the optimisation problems, we show intractability for most interesting cases. Particularly, in contrast to the tractability of generalised reachability in the case with singleton target sets, the optimisation problem is coNP-hard when Eve tries to maximise the number of target sets that are visited. Tractability of this case can be recovered in a different optimisation setting where Eve is required to pledge a maximum sized subset of target sets that she can guarantee to visit.
