A greedy heuristic for graph burning

Abstract

Given a graph $G$, the optimization version of the graph burning problem seeks for a sequence of vertices, $(u_1,u_2,...,u_p) \in V(G)^p$, with minimum $p$ and such that every $v \in V(G)$ has distance at most $p-i$ to some vertex $u_i$. The length $p$ of the optimal solution is known as the burning number and is denoted by $b(G)$, an invariant that helps quantify the graph's vulnerability to contagion. This paper explores the advantages and limitations of an $\mathcal{O}(mn + pn^2)$ deterministic greedy heuristic for this problem, where $n$ is the graph's order, $m$ is the graph's size, and $p$ is a guess on $b(G)$. This heuristic is based on the relationship between the graph burning problem and the clustered maximum coverage problem, and despite having limitations on paths and cycles, it found most of the optimal and best-known solutions of benchmark and synthetic graphs with up to 102400 vertices. Beyond practical advantages, our work unveils some of the fundamental aspects of graph burning: its relationship with a generalization of a classical coverage problem and compact integer programs. With this knowledge, better algorithms might be designed in the future.