Approximation Bounds for Black Hole Search Problems

A black hole is a highly harmful stationary process residing in a node of a network and destroying all mobile agents visiting the node without leaving any trace. The Black Hole Search is the task of locating all black holes in a network, through the exploration of its nodes by a set of mobile agents. In this article we consider the problem of designing the fastest Black Hole Search, given the map of the network, the starting node and a subset of nodes of the network initially known to be safe. We study the version of this problem that assumes that there is at most one black hole in the network and there are two agents, which move in synchronized steps. We prove that this problem is not polynomial-time approximable within any constant factor less than 398/388 (unless P = NP). We give a 6-approximation algorithm, thus improving on the 9.3-approximation algorithm from (Czyzowicz et al., Fundamenta Informaticae 71 (2006), 229-242). We also prove APX-hardness for a restricted version of the problem, which only the starting node is initially known to be safe. (C) 2008 Wiley Periodicals, Inc. NETWORKS, Vol. 52(4), 216-226 2008