On the Complexity of Kings

Abstract

A k-king in a directed graph is a node from which each node in the graph can be reached via paths of length at most k. Recently, kings have proven useful in theoretical computer science, in particular in the study of the complexity of reachability problems and semifeasible sets. In this paper, we study the complexity of recognizing k-kings. For each succinctly specified family of tournaments (completely oriented digraphs), the k-king problem is easily seen to belong to $\Pi_2^{\mathrm p}$. We prove that the complexity of kingship problems is a rich enough vocabulary to pinpoint every nontrivial many-one degree in $\Pi_2^{\mathrm p}$. That is, we show that for every k ≥ 2 every set in $\Pi_2^{\mathrm p}$ other than ∅ and Σ ^* is equivalent to a k-king problem under $\leq_{\mathrm m}^{\mathrm p}$-reductions. The equivalence can be instantiated via a simple padding function. Our results can be used to show that the radius problem for arbitrary succinctly represented graphs is $\Sigma_3^{\rm p}$-complete. In contrast, the diameter problem for arbitrary succinctly represented graphs (or even tournaments) is $\Pi_2^{\mathrm p}$-complete.