Approximation Algorithms for the Graph Burning on Cactus and Directed Trees

Given a graph $G=(V, E)$, the problem of Graph Burning is to find a sequence of nodes from $V$, called a burning sequence, to burn the whole graph. This is a discrete-step process, and at each step, an unburned vertex is selected as an agent to spread fire to its neighbors by marking it as a burnt node. A burnt node spreads the fire to its neighbors at the next consecutive step. The goal is to find the burning sequence of minimum length. The Graph Burning 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. The Graph Burning on directed graphs is more challenging than on undirected graphs. In this paper, we propose 1) A $2.75$-approximation algorithm for a cactus graph (undirected), 2) A $3$-approximation algorithm for multi-rooted directed trees (polytree) and 3) A $1.905$-approximation algorithm for single-rooted directed tree (arborescence). We implement all the three approximation algorithms and the results are shown for randomly generated cactus graphs and directed trees.