Complexity of Unambiguous Problems in $Σ^P_2$ (2510.19084v1)

Abstract: The complexity class $\bf{\Sigma^P_2}$ comprises problems based on polynomial-time checkable binary relations $\phi(x,y)$ in which we ask whether there exists $x$ such that for all $y$, $\phi(x,y)$ holds. We let $\bf{U\Sigma^P_2}$ denote the subclass of unambiguous problems in $\bf{\Sigma^P_2}$, namely those whose yes-instances correspond with a unique choice of $x$. $\bf{U\Sigma^P_2}$ is unlikely to have complete problems, but we identify various syntactic subclasses associated with general properties of $\phi$ that guarantee uniqueness. We use these to classify the complexity of problems arising in social choice and game theory, such as 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. We classify these problems, showing the first is $\bf{\Delta^P_2}$-complete, the second and third are complete for a class we term $\bf{PCW}$ (Polynomial Condorcet Winner), and the fourth for a class we term $\bf{PTW}$ (Polynomial Tournament Winner). We define another unambiguous class, $\bf{PMA}$ (Polynomial Majority Argument), seemingly incomparable to $\bf{PTW}$ and $\bf{PCW}$. We show that with randomization, $\bf{PCW}$ and $\bf{PTW}$ coincide with $\bf{\Delta^P_2}$, and $\bf{PMA}$ is contained in $\bf{\Delta^P_2}$. Specifically, we prove: $\bf{\Delta^P_2} \subseteq \bf{PCW} \subseteq \bf{PTW} \subseteq \bf{S^P_2}$, and $\bf{coNP} \subseteq \bf{PMA} \subseteq \bf{S^P_2}$ (and it is known that $\bf{S^P_2}\subseteq \bf{ZPP^{NP}} \subseteq \bf{\Sigma^P_2} \cap \bf{\Pi^P_2}$). We demonstrate that unambiguity can substantially reduce computational complexity by considering ambiguous variants of our problems, and showing they are $\bf{\Sigma^P_2}$-complete. Finally, we study the unambiguous problem of finding a weakly dominant strategy in a game, which seems not to lie in $\bf{\Sigma^P_2}$.