图燃烧的一种贪心启发式算法
A greedy heuristic for graph burning
AI总结：
本文探讨了一种基于聚类最大覆盖问题的复杂度为 $\mathcal{O}(mn + pn^2)$ 的确定性贪心启发式算法用于解决图燃烧问题，揭示了该问题的基本性质，并在多达 102400 个顶点的图中找到了大多数最优解。
摘要：
给定图 $G$，图燃烧问题的优化版本旨在寻找一个顶点序列 $(u_1,u_2,...,u_p) \in V(G)^p$，使得 $p$ 最小，且每个 $v \in V(G)$ 到某个顶点 $u_i$ 的距离至多为 $p-i$。最优解的长度 $p$ 被称为燃烧数，记为 $b(G)$，这是一个有助于量化图对传染脆弱性的不变量。本文探讨了一种复杂度为 $\mathcal{O}(mn + pn^2)$ 的确定性贪心启发式算法在解决此问题时的优势与局限性，其中 $n$ 为图的阶数，$m$ 为图的边数，$p$ 是对 $b(G)$ 的猜测值。该启发式算法基于图燃烧问题与聚类最大覆盖问题之间的关系，尽管在路径和环上存在局限性，但它在多达 102400 个顶点的基准图和合成图中找到了大多数最优解和已知最佳解。除了实际优势外，我们的工作还揭示了图燃烧的一些基本方面：它与经典覆盖问题推广形式的关系以及紧凑整数规划。凭借这些知识，未来可能会设计出更好的算法。
英文摘要：
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.
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