Firefighting on Trees: (1 − 1/e)–Approximation, Fixed Parameter Tractability and a Subexponential Algorithm

Abstract

The firefighter problem is defined as follows. Initially, a fire breaks out at a vertex r of a graph G. In each subsequent time unit, a firefighter chooses a vertex not yet on fire and protects it, and the fire spreads to all unprotected neighbors of the vertices on fire. The objective is to choose a sequence of vertices for the firefighter to protect so as to save the maximum number of vertices. The firefighter problem can be used to model the spread of fire, diseases, computer viruses and suchlike in a macro-control level. In this paper, we study algorithmic aspects of the firefighter problem on trees, which is NP-hard even for trees of maximum degree 3. We present a (1 - 1/e)-approximation algorithm based on LP relaxation and randomized rounding, and give several FPT algorithms using a random separation technique of Cai, Chan and Chan. Furthermore, we obtain an $2^{O(\sqrt{n}\log n)}$-time subexponential algorithm.