Monotonicity of Non-deterministic Graph Searching

Abstract

In graph searching, a team of searchers is aiming at capturing a fugitive moving in a graph. In the initial variant, called invisible graph searching, the searchers do not know the position of the fugitive until they catch it. In another variant, the searchers know the position of the fugitive, i.e. the fugitive is visible. This latter variant is called visible graph searching. A search strategy that catches any fugitive in such a way that, the part of the graph reachable by the fugitive never grows is called monotone. A priori, monotone strategies may require more searchers than general strategies to catch any fugitive. This is however not the case for visible and invisible graph searching. Two important consequences of the monotonicity of visible and invisible graph searching are: (1) the decision problem corresponding to the computation of the smallest number of searchers required to clear a graph is in NP, and (2) computing optimal search strategies is simplified by taking into account that there exist some that never backtrack.

Fomin et al. (2005) introduced an important graph searching variant, called non-deterministic graph searching, that unifies visible and invisible graph searching. In this variant, the fugitive is invisible, and the searchers can query an oracle that knows the current position of the fugitive. The question of the monotonicity of non-deterministic graph searching is however left open.

In this paper, we prove that non-deterministic graph searching is monotone. In particular, this result is a unified proof of monotonicity for visible and invisible graph searching. As a consequence, the decision problem corresponding to non-determinisitic graph searching belongs to NP. Moreover, the exact algorithms designed by Fomin et al. do compute optimal non-deterministic search strategies.

The second author received additional supports from the project “Alpage” of the ACI Masses de Données, from the project “Fragile” of the ACI Sécurité Informatique, and from the project “Grand Large” of INRIA.