Determining a Slater Winner is Complete for Parallel Access to NP

Abstract

We consider the complexity of deciding the winner of an election under the Slater rule. In this setting we are given a tournament $T = (V, A)$, where the vertices of V represent candidates and the direction of each arc indicates which of the two endpoints is preferable for the majority of voters. The Slater score of a vertex $v\in V$ is defined as the minimum number of arcs that need to be reversed so that $T$ becomes acyclic and $v$ becomes the winner. We say that $v$ is a Slater winner in $T$ if $v$ has minimum Slater score in $T$. Deciding if a vertex is a Slater winner in a tournament has long been known to be NP-hard. However, the best known complexity upper bound for this problem is the class $\Theta_2^p$, which corresponds to polynomial-time Turing machines with parallel access to an NP oracle. In this paper we close this gap by showing that the problem is $\Theta_2^p$-complete, and that this hardness applies to instances constructible by aggregating the preferences of 7 voters.