Complexity of Unambiguous Problems in $\Sigma^P_2$

TL;DR

This paper explores the complexity of unambiguous problems in the class ^P_2, introducing subclasses with guaranteed uniqueness properties, and establishes their complexity bounds, resolving open questions in the field.

Contribution

The paper defines three subclasses of unambiguous ^P_2 problems, analyzes their complexity bounds, and applies this framework to practical problems, including resolving open questions.

Findings

All subclasses are contained in S_2^P, making them easier than ^P_2.

Characterized the complexity of practical problems like dominance in games and winners in tournaments.

Resolved an open question about strong-popularity in additive hedonic games.

Abstract

Various practical problems within the class $\Sigma_{2}^P$ possess an unambiguity property, meaning that yes-instances correspond with a unique witness. The semantic class containing all unambiguous $\Sigma_{2}^P$ problems is denoted $U\Sigma_{2}^P$. 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 $U\Sigma_{2}^P$ 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 $\Sigma_{2}^P$ 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.