Improved Approximation Algorithm for Graph Burning on Trees
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Given a graph G=(V,E), the problem of is to find a sequence of nodes from V, called burning sequence, in order to burn the whole graph. This is a discrete-step process, in each step an unburned vertex is selected as an agent to spread fire to its neighbors by marking it as a burnt node. A node that is burnt spreads the fire to its neighbors at the next consecutive step. The goal is to find the burning sequence of minimum length. The problem is NP-Hard for general graphs and even for binary trees. A few approximation results are known, including a 3-approximation algorithm for general graphs and a 2- approximation algorithm for trees. In this paper, we propose an approximation algorithm for trees that produces a burning sequence of length at most ⌊ 1.75b(T) ⌋ + 1, where b(T) is length of the optimal burning sequence, also called the burning number of the tree T. In other words, we achieve an approximation factor of (⌊ 1.75b(T) ⌋ + 1)/b(T).
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