Monotonicity in digraph search problems

  • Authors:

  • Boting Yang;Yi Cao

  • Affiliations:

  • Department of Computer Science, University of Regina, Regina, Saskatchewan, S4S 0A2, Canada;Department of Computer Science, University of Regina, Regina, Saskatchewan, S4S 0A2, Canada

  • Venue:
  • Theoretical Computer Science
  • Year:
  • 2008

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Abstract

In this paper, we study the monotonicity and complexity of five digraph search problems: directed searching, mixed directed searching, internal directed searching, internal strong searching, and internal weak searching. In the first three search problems, both searchers and intruder must follow the edge directions when they move along edges. In the internal strong search problem, the intruder must move in the edge directions but searchers need not. In the internal weak search problem, searchers must move in the edge directions but the intruder need not. There are three actions for searchers in the first two search problems: placing, removing and sliding, and there are only two actions for searchers in the last three internal search problems: placing and sliding. Note that the internal strong searching is a ''strong'' version of the internal directed searching, the internal weak searching is a ''weak'' version of the internal directed searching, and the internal edge searching is an analogy of the internal directed searching on undirected graphs. We prove that the first three problems are monotonic and the last two problems are non-monotonic, respectively. It is interesting that the internal directed searching is monotonic while the internal strong searching, the internal weak searching and the internal edge searching are all non-monotonic. We also show that the first four problems are NP-complete and the last problem is NP-hard. We solve the open problem on whether a non-monotonic searching problem can be NP-complete.