Complexity of Unambiguous Problems in $Σ^P_2$

TL;DR

The paper investigates unambiguous problems at the second level of the polynomial hierarchy, defining $UΣ^P_2$ and introducing three natural unambiguous classes—$PTW$, $PCW$, and $PMA$—to capture tournament- and majority-based structures. It provides precise classifications for social-choice and game-theoretic tasks: a dominating strategy is $Δ^P_2$-complete, Condorcet-winner and strongly popular partitions are $PCW$-complete, and winner problems in tournaments are in $PTW$, with randomization collapsing several hierarchies toward $Δ^P_2$; a separate class $PMA$ sits in $S_2^P$ with $coNP subseteq PMA$ (though it contains $coNP$). The authors also show that relaxing unambiguity yields $ ext{Σ}^P_2$-complete problems and analyze a weak-dominant strategy problem that appears to lie outside $Σ^P_2$, illustrating the nuanced impact of unambiguity on computational hardness. Overall, the work provides a cohesive taxonomy linking unambiguous $Σ^P_2$ problems to classical complexity classes and to practical problems in social choice and game theory, and it highlights both the power and limits of unambiguity in structural decision problems.

Abstract

The complexity class $\bf{Σ^P_2}$ comprises problems based on polynomial-time checkable binary relations $φ(x,y)$ in which we ask whether there exists $x$ such that for all $y$, $φ(x,y)$ holds. We let $\bf{UΣ^P_2}$ denote the subclass of unambiguous problems in $\bf{Σ^P_2}$, namely those whose yes-instances correspond with a unique choice of $x$. $\bf{UΣ^P_2}$ is unlikely to have complete problems, but we identify various syntactic subclasses associated with general properties of $φ$ 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 $\bf{Δ^P_2}$-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 $\bf{Δ^P_2}$, and $\bf{PMA}$ is contained in $\bf{Δ^P_2}$. Specifically, we prove: $\bf{Δ^P_2} \subseteq \bf{PCW} \subseteq \bf{PTW} \subseteq \bf{S^P_2}$, and $\bf{coNP} \subseteq \bf{PMA} \subseteq \bf{S^P_2}$ (and it is known that $\bf{S^P_2}\subseteq \bf{ZPP^{NP}} \subseteq \bf{Σ^P_2} \cap \bf{Π^P_2}$). We demonstrate that unambiguity can substantially reduce computational complexity by considering ambiguous variants of our problems, and showing they are $\bf{Σ^P_2}$-complete. Finally, we study the unambiguous problem of finding a weakly dominant strategy in a game, which seems not to lie in $\bf{Σ^P_2}$.

Figures (6)

• Figure 1: Our proposed complexity classes (in red) relative to known subclasses of $\bf{\Sigma_{2}^P}$.
• Figure 2: Assignment gadget of bit $\mathtt{x}_i$. Dots represent single agents; circles represent sets of $m$ replicas.
• Figure 3: Copy-gadget of gate $\mathcal{G}_i=\mathtt{x}_i$.
• Figure 4: Not-gadget of gate $\mathcal{G}_i=\neg\mathcal{G}_j$.
• Figure 5: And-gadget of gate $\mathcal{G}_i=\mathcal{G}_j\land \mathcal{G}_k$.

Theorems & Definitions (158)

• Definition 2.1
• Definition 2.2
• Definition 3.1
• Theorem 3.2
• Theorem 3.3
• Theorem 3.4
• Definition 4.1
• Theorem 4.2
• Theorem 4.3
• Theorem 4.4