Determining a Slater Winner is Complete for Parallel Access to NP
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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∈ 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 Θ_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 Θ_2^p-complete, and that this hardness applies to instances constructible by aggregating the preferences of 7 voters.
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