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The Complexity of Kings

Edith Hemaspaandra·Lane A. Hemaspaandra·Osamu Watanabe
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Abstract

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 $Π_2^p$ [HOZZ]. We prove that this bound is optimal: We construct a succinctly specified tournament family whose king problem is $Π_2^p$-complete. It follows easily from our proof approach that the problem of testing kingship in succinctly specified graphs (which need not be tournaments) is $Π_2^p$-complete. We also obtain $Π_2^p$-completeness results for k-kings in succinctly specified j-partite tournaments, $k,j \geq 2$, and we generalize our main construction to show that $Π_2^p$-completeness holds for testing k-kingship in succinctly specified families of tournaments for all $k \geq 2$.

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Edith Hemaspaandra, Lane A. Hemaspaandra, Osamu Watanabe. 2005-06-14. The Complexity of Kings. https://arxiv.org/abs/cs/0506055

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