Various practical problems within the class $Σ_{2}^P$ possess an unambiguity property, meaning that yes-instances correspond with a unique witness. The semantic class containing all unambiguous $Σ_{2}^P$ problems is denoted $UΣ_{2}^P$. Examples include the existence of (1) a dominating strategy in a game, (2) a Condorcet winner, (3) a strongly popular partition in hedonic games, and (4) a winner (source) in a tournament. The computational complexity of unambiguous problems is not well understood, leaving many questions unresolved. We address this gap in a broad complexity-theoretic sense; our main contributions consist of the following. - We identify three syntactic subclasses of $UΣ_{2}^P$ associated with general properties of problems that guarantee uniqueness: Polynomial Tournament Winner (PTW), Polynomial Condorcet Winner (PCW), and Polynomial Majority Argument (PMA). - We establish complexity upper and lower bounds for our proposed classes. In particular, we show that they are all contained in $S_2^P$ and are thus significantly easier than the immediate $Σ_{2}^P$ upper bound. - We characterize the complexity of various practical problems using this framework. In particular, we resolve an open question by Brandt and Bullinger (JAIR '22) and Bullinger and Gilboa (IJCAI '25) concerning strong-popularity in additive hedonic games.

翻译：属于$Σ_{2}^P$类的各种实际问题具有无歧义性，即肯定实例对应唯一的见证。包含所有无歧义$Σ_{2}^P$问题的语义类记为$UΣ_{2}^P$。例如包括：(1) 博弈中的主导策略存在性，(2) Condorcet赢家的存在性，(3) 享乐博弈中强流行分割的存在性，以及(4) 锦标赛中赢家（源点）的存在性。无歧义问题的计算复杂性尚未得到充分理解，许多问题悬而未决。我们以广泛的复杂性理论视角填补这一空白；主要贡献如下： - 我们识别出$UΣ_{2}^P$的三种语法子类，它们与保证唯一性的一般问题性质相关联：多项式锦标赛赢家(PTW)、多项式Condorcet赢家(PCW)和多项式多数论证(PMA)。 - 我们为所提出的类建立了复杂性上下界。特别地，我们证明它们都包含在$S_2^P$中，因此比直接的$Σ_{2}^P$上界显著更容易。 - 我们利用这一框架刻画了各种实际问题的复杂性。特别地，我们解决了Brandt和Bullinger (JAIR '22)以及Bullinger和Gilboa (IJCAI '25)关于加性享乐博弈中强流行性的开放问题。