The Complexity of Kings

A king in a directed graph is a node from which each node in the graph can be reached via 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 [Lan53]. Recently, kings have proven useful in theoretical computer science, in particular in the study of the complexity of the semifeasible sets [HNP98,HT05] and in the study of the complexity of reachability problems [Tan01,NT02]. In this paper, we study the complexity of recognizing kings. For each succinctly specified family of tournaments, the king problem is known to belong to Πp2 [HOZZ]. We prove that this bound is optimal: We construct a succinctly specified tournament family whose king problem is Πp2-complete. It follows easily from our proof approach that the problem of testing kingship in succinctly specified graphs (which need not be tournaments) is Πp2-complete. We also obtain Π p 2-completeness results for k-kings in succinctly specified j-partite tournaments, k, j ≥ 2, and we generalize our main construction to show that Πp2-completeness holds for testing k-kingship in succinctly specified families of tournaments for all k ≥ 2. ∗This report will appear as a technical/research report at U.R., Tokyo Tech, and the Computing Research Repository. †Supported in part by grant NSF-CCR-0311021. Work done in part while on sabbatical at the University of Rochester and while visiting the Tokyo Institute of Technology. ‡Supported in part by grant NSF-CCF-0426761 and JSPS Invitational Fellowship S-05022. Work done in part while visiting the Tokyo Institute of Technology. §Supported in part by “New Horizons in Computing” (2004–2006), an MEXT Grant-in-Aid for Scientific Research on Priority Areas.