Hardness and approximation results for black hole search in arbitrary graphs

Authors:
Ralf Klasing;Euripides Markou;Tomasz Radzik;Fabiano Sarracco
Affiliations:
MASCOTTE project, I3S-CNRS/INRIA/Université de Nice-Sophia Antipolis, Sophia Antipolis Cedex, France;Department of Informatics and Telecommunications, National and Kapodistrian University of Athens;Department of Computer Science, King's College London, London, UK;Dipartimento di Informatica e Sistemistica, Università di Roma “La Sapienza”
Venue:
SIROCCO'05 Proceedings of the 12th international conference on Structural Information and Communication Complexity
Year:
2005

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Abstract

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. We consider the task of locating a black hole in a (partially) synchronous arbitrary network, assuming an upper bound on the time of any edge traversal by an agent. For a given graph and a given starting node we are interested in finding the fastest possible Black Hole Search by two agents (the minimum number of agents capable to identify a black hole). We prove that this problem is NP-hard in arbitrary graphs, thus solving an open problem stated in [2]. We also give a 7/2-approximation algorithm, thus improving on the 4-approximation scheme observed in [2]. Our approach is to explore the given input graph via some spanning tree. Even if it represents a very natural technique, we prove that this approach cannot achieve an approximation ratio better than 3/2.