Improved Approximation Algorithms for Firefighter Problem on Trees

概要

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The firefighter problem is used to model the spread of fire, infectious diseases, and computer viruses. This paper deals with firefighter problem on rooted trees. It is known that the firefighter problem is NP-hard even for rooted trees of maximum degree 3. We propose techniques to improve a given approximation algorithm. First, we introduce an implicit enumeration technique. By applying the technique to existing (1-1/e)-approximation algorithm, we obtain $(1- {k-1 \\over (k-1)e + 1})$-approximation algorithm when a root has k children. In case of ternary trees, k=3 and thus the approximation ratio satisfies $(1- {k-1 \\over (k-1)e + 1})$ ≥ 0.6892, which improves the existing result 1-1/e ≥ 0.6321. Second technique is based on backward induction and improves an approximation algorithm for firefighter problem on ternary trees. If we apply the technique to existing (1-1/e)-approximation algorithm, we obtain 0.6976-approximation algorithm. Lastly, we combine the above two techniques and obtain 0.7144-approximation algorithm for firefighter problem on ternary trees.