A greedy heuristic for graph burning

TL;DR

The paper tackles the graph burning problem (GBP), which seeks a minimum-length burning sequence with burning number $b(G)$. It proposes a deterministic greedy heuristic Gr by reducing GBP to a sequence of clustered maximum coverage problems (CMCP) using a guess $p$ on $b(G)$ and solving via a $O(mn+pn^2)$ process, with binary search to identify $b(G)$. A key theoretical contribution is a $1/2$-approximation for CMCP, which is then adapted into the GBP heuristic, plus an enhanced variant GrP that repeats Gr from multiple starts. Empirically, Gr and GrP perform well on real-world and synthetic graphs, often matching or approaching optimal solutions and surpassing previous ILP-based formulations in some cases, while clarifying the structural link between GBP and CMCP for future algorithm design. The work provides a practical, scalable framework for estimating GBP solutions and offers a foundation for developing more advanced GBP algorithms leveraging CMCP insights.

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.

Figures (6)

• Figure 1: Burning process of the optimal burning sequence $(v_4,v_8,v_1)$; notice that $b(G)=3$. At each discrete time step, the fire propagates to the neighbors of the previously burned vertices, and the next vertex in the sequence gets burned too. In the end, all vertices are burned.
• Figure 2: In this instance for CMCP, $p=6$, $\mathcal{C}_1 = \{\mathcal{S}_{1,1},\mathcal{S}_{1,2}\}$, $\mathcal{C}_2 = \{\mathcal{S}_{2,1},\mathcal{S}_{2,2}\}$, $\mathcal{C}_3 = \{\mathcal{S}_{3,1},\mathcal{S}_{3,2}\}$, $\mathcal{C}_4 = \{\mathcal{S}_{4,1}\}$, $\mathcal{C}_5 = \{\mathcal{S}_{5,1}\}$, and $\mathcal{C}_6 = \{\mathcal{S}_{6,1}\}$. The optimal solution $\{\mathcal{S}_{1,1}, \mathcal{S}_{2,1}, \mathcal{S}_{3,1}, \mathcal{S}_{4,1}, \mathcal{S}_{5,1}, \mathcal{S}_{6,1}\}$ covers $OPT=30$ squares. If the greedy approximation algorithm selects $\mathcal{S}_{1,2}$ at the first iteration, all the squares inside $\mathcal{S}_{1,1}$ get blocked because no other subsets from the remaining clusters cover them. If the algorithm selects $\mathcal{S}_{2,2}$ in the second iteration, all the squares inside $\mathcal{S}_{2,1}$ get blocked. Finally, if the algorithm selects $\mathcal{S}_{3,2}$ at the third iteration, all the squares inside $\mathcal{S}_{3,1}$ get blocked. Thus, the next three selected subsets won't be able to cover any uncovered squares and the returned solution will be $\{\mathcal{S}_{1,2},\mathcal{S}_{2,2},\mathcal{S}_{3,2},\mathcal{S}_{4,1},\mathcal{S}_{5,1},\mathcal{S}_{6,1}\}$, which covers $OPT/2=15$ squares.
• Figure 3: An instance $\Pi_\alpha$ with $p=4$ and $OPT(\Pi_\alpha)=10$.
• Figure 4: An instance $\Pi_\beta^\prime$ with $p=3$ and $OPT(\Pi_\beta^\prime)=9$. $\Pi_\beta^\prime$ results from removing $\mathcal{C}_2$ from $\Pi_\alpha$.
• Figure 5: An instance $\Pi_\beta$ with $p=3$ and $OPT(\Pi_\beta)=7$. $\Pi_\beta$ results from removing some specific elements from $\Pi_\beta^\prime$.

Theorems & Definitions (28)

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