On the complexity of kings

A king in a directed graph is a vertex from which each vertex in the graph can be reached through paths of length at most two. There is a broad literature on tournaments (completely oriented digraphs), and it has been known for more than half a century that all tournaments have at least one king. 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 article, we study the complexity of recognizing kings. For each succinctly specified family of tournaments, the king problem is already known to belong to Π₂^p. We prove that the complexity of kingship problems is a rich enough vocabulary to pinpoint every nontrivial many-one degree in Π₂^p. That is, we show that every set in Π₂^p other than 0{combining long solidus overlay} and Σ^* is equivalent to a king problem under ≤_m^p-reductions. Indeed, we show that the equivalence can even be realized by relatively simple padding, and holds even if the notion of kings is redefined to refer to k-kings (for any fixed k ≥ 2)-vertices from which all vertices can be reached through paths of length at most k. In contrast, we prove that for each succinctly specified family of tournaments the source problem (the problem of deciding whether a given vertex v has the property that there exists a k such that v is a k-king) also falls within Π₂^p, yet cannot be Π₂^p-complete-or even NP-hard-unless P = NP. Using these and related techniques, we obtain a broad range of additional results about the complexity of king problems, diameter problems, and radius problems. It follows easily from our proof approach that the problem of testing kingship in succinctly specified graphs (which need not be tournaments) is Π₂^p-complete. We show that the radius problem for arbitrary succinctly represented graphs is Σ₃^p-complete, but that in contrast the diameter problem for arbitrary succinctly represented graphs (or even tournaments) is Π₂^p-complete.