We propose an exact algorithm for the Graph Burning Problem ($\texttt{GBP}$), an NP-hard optimization problem that models the spread of influence on social networks. Given a graph $G$ with vertex set $V$, the objective is to find a sequence of $k$ vertices in $V$, namely, $v_1, v_2, \dots, v_k$, such that $k$ is minimum and $\bigcup_{i = 1}^{k} \{u\! \in\! V\! : d(u, v_i) \leq k - i\} = V$, where $d(u,v)$ denotes the distance between $u$ and $v$. We formulate the problem as a set covering integer programming model and design a row generation algorithm for the $\texttt{GBP}$. Our method exploits the fact that a very small number of covering constraints is often sufficient for solving the integer model, allowing the corresponding rows to be generated on demand. To date, the most efficient exact algorithm for the $\texttt{GBP}$, denoted here by $\texttt{GDCA}$, is able to obtain optimal solutions for graphs with up to 14,000 vertices within two hours of execution. In comparison, our algorithm finds provably optimal solutions approximately 236 times faster, on average, than $\texttt{GDCA}$. For larger graphs, memory space becomes a limiting factor for $\texttt{GDCA}$. Our algorithm, however, solves real-world instances with almost 200,000 vertices in less than 35 seconds, increasing the size of graphs for which optimal solutions are known by a factor of 14.

翻译：本文针对图燃烧问题（$\texttt{GBP}$）提出了一种精确算法，该问题是一个用于模拟社交网络影响力传播的NP难优化问题。给定一个顶点集为$V$的图$G$，其目标是在$V$中找到一个包含$k$个顶点的序列$v_1, v_2, \dots, v_k$，使得$k$值最小且满足$\bigcup_{i = 1}^{k} \{u\! \in\! V\! : d(u, v_i) \leq k - i\} = V$，其中$d(u,v)$表示顶点$u$与$v$之间的距离。我们将该问题建模为集合覆盖整数规划模型，并为$\texttt{GBP}$设计了一种行生成算法。我们的方法利用了以下事实：通常只需极少量的覆盖约束即可求解该整数模型，从而允许按需生成对应的约束行。迄今为止，最有效的$\texttt{GBP}$精确算法（本文记为$\texttt{GDCA}$）能够在两小时执行时间内获得最多14,000个顶点图的最优解。相比之下，我们的算法求解可证明最优解的速度平均比$\texttt{GDCA}$快约236倍。对于更大规模的图，内存空间成为$\texttt{GDCA}$的限制因素。然而，我们的算法在35秒内即可求解包含近200,000个顶点的实际实例，将已知最优解可求解的图规模提升了14倍。