On the complexity of kings

  • Authors:

  • Edith Hemaspaandra;Lane A. Hemaspaandra;Till Tantau;Osamu Watanabe

  • Affiliations:

  • Dept. of Comput. Science, Rochester Institute of Technology, Rochester, NY;Dept. of Comput. Science, University of Rochester, Rochester, NY;Inst. of Theoretical Comput. Science, Universität zu Lübeck, Germany;Dept. of Math. & Comput. Sciences, Tokyo Institute of Technology, Japan

  • Venue:
  • FCT'07 Proceedings of the 16th international conference on Fundamentals of Computation Theory
  • Year:
  • 2007

Quantified Score

Hi-index 0.00

Visualization

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 Π2P. We prove that the complexity of kingship problems is a rich enough vocabulary to pinpoint every nontrivial many-one degree in Π2P. That is, we show that for every k ≥ 2 every set in Π2P other than θ and Σ* is equivalent to a k-king problem under ≤mp-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 Σ3P-complete. In contrast, the diameter problem for arbitrary succinctly represented graphs (or even tournaments) is Π2P2 -complete.