The Complexity of Kings

TL;DR

This paper investigates the computational complexity of recognizing kings in directed graphs, proving that the problem is a2_2^p-complete for various classes of succinctly specified tournaments, establishing optimal bounds.

Contribution

It demonstrates that the king recognition problem is a2_2^p-complete for succinctly specified tournaments and extends this result to k-kings and non-tournament graphs.

Findings

King recognition is a2_2^p-complete in succinctly specified tournaments.

The a2_2^p-completeness extends to k-kings and non-tournament graphs.

The complexity bounds are shown to be optimal.

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 $\Pi_2^p$ [HOZZ]. We prove that this bound is optimal: We construct a succinctly specified tournament family whose king problem is $\Pi_2^p$-complete. It follows easily from our proof approach that the problem of testing kingship in succinctly specified graphs (which need not be tournaments) is $\Pi_2^p$-complete. We also obtain $\Pi_2^p$-completeness results for k-kings in succinctly specified j-partite tournaments, $k,j \geq 2$, and we generalize our main construction to show that $\Pi_2^p$-completeness holds for testing k-kingship in succinctly specified families of tournaments for all $k \geq 2$.