Consider an information diffusion process on a graph $G$ that starts with $k>0$ burnt vertices, and at each subsequent step, burns the neighbors of the currently burnt vertices, as well as $k$ other unburnt vertices. The \emph{$k$-burning number} of $G$ is the minimum number of steps $b_k(G)$ such that all the vertices can be burned within $b_k(G)$ steps. Note that the last step may have smaller than $k$ unburnt vertices available, where all of them are burned. The $1$-burning number coincides with the well-known burning number problem, which was proposed to model the spread of social contagion. The generalization to $k$-burning number allows us to examine different worst-case contagion scenarios by varying the spread factor $k$. In this paper we prove that computing $k$-burning number is APX-hard, for any fixed constant $k$. We then give an $O((n+m)\log n)$-time 3-approximation algorithm for computing $k$-burning number, for any $k\ge 1$, where $n$ and $m$ are the number of vertices and edges, respectively. Finally, we show that even if the burning sources are given as an input, computing a burning sequence itself is an NP-hard problem.

中文翻译：

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k-燃烧数问题的 APX 硬度和近似

考虑图 $G$ 上的信息扩散过程，该过程以 $k>0$ 个已烧毁的顶点开始，并且在随后的每个步骤中，烧毁当前烧毁顶点的邻居，以及 $k$ 其他未烧毁的顶点。$G$ 的\emph{$k$-burning number} 是$b_k(G)$ 的最小步数，使得所有顶点都可以在$b_k(G)$ 步内燃烧。请注意，最后一步可能有小于 $k$ 的未燃烧顶点可用，所有这些顶点都被烧毁。1 美元的燃烧数字与众所周知的燃烧数字问题相吻合，该问题被提议用于模拟社会传染的传播。对 $k$-burning 数的概括允许我们通过改变传播因子 $k$ 来检查不同的最坏情况传染场景。在本文中，我们证明对于任何固定常数 $k$，计算 $k$-burning 数是 APX-hard 的。然后我们给出一个 $O((n+m)\log n)$-time 3-approximation 算法来计算 $k$-burning 数，对于任何 $k\ge 1$，其中 $n$ 和 $m$ 是分别为顶点数和边数。最后，我们表明即使燃烧源作为输入给出，计算燃烧序列本身也是一个 NP-hard 问题。