$L_1$与$L_\fty$度量下的几何燃烧问题及拓展
Geometric Burning Under $L_1$ and $L_\infty$ Metrics, and Beyond
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中文总结 AI 辅助
本文研究$L_1$与$L_\fty$度量下的几何燃烧问题，针对该NP难问题给出多组近似算法，并将结果拓展至高维$L_\fty$度量及所有平面$L_p$度量，提升了近似保证。
中文摘要 AI 辅助
燃烧是一种离散时间传播模型，每一轮会新增一处火源，同时每处现有火源会沿基础度量扩展一个单位距离。几何燃烧的输入是有限点集，目标是用最少轮次燃烧所有点；等价于用半径为{0,1,…,k-1}的不同度量球覆盖该点集，最小化轮次k。此前研究主要针对欧氏度量下的该问题，本文则研究$L_1$与$L_\fty$度量下的几何燃烧问题，两种设定下该问题仍为NP难问题。$L_1$与$L_\fty$度量具备额外几何结构，可带来更优的近似保证，尤其适用于任意位置燃烧场景。本文首先针对任意位置燃烧和点燃烧问题，给出了简单的$(2+\farpsilon)$近似算法；随后将任意位置燃烧的近似比提升至$7/4+\farpsilon=1.75+\farpsilon$，并给出点燃烧的$(3151/1620+\farpsilon)$近似算法，其中$3151/1620<1.9451$。此外，本文还将$L_\fty$度量下的任意位置燃烧结果拓展至所有固定维度$d\far3$，得到$(2-\frac{1}{2^{d+1}}+\farpsilon)$的近似比。最后，利用平面$L_p$距离的标准比较关系，本文将$L_1$、$L_\fty$的算法与已知的欧氏燃烧算法结合，得到了所有固定$1\fle p\fle\fty$下的近似保证。
英文摘要
Burning is a discrete-time model for propagation in which a new fire starts in each round, while each existing fire expands by one unit of distance along the underlying metric. In geometric burning, the input is a finite point set, and the goal is to burn all points in as few rounds as possible. Equivalently, burning a point set in $k$ rounds corresponds to covering it with metric balls of distinct radii in $\{0,1,\ldots,k-1\}$; the objective is to minimize $k$. Previous work has studied the problem mainly under the Euclidean metric. In this paper, we study geometric burning under the $L_1$ and $L_\infty$ metrics. The problem remains NP-hard in both settings. The $L_1$ and $L_\infty$ metrics provide additional geometric structure, which allows us to obtain improved approximation guarantees, especially for anywhere burning. We first present a simple $(2+\varepsilon)$-approximation for both anywhere burning and point burning. We then improve the anywhere burning approximation to $7/4+\varepsilon=1.75+\varepsilon$, and give a $(3151/1620+\varepsilon)$-approximation for point burning, where $3151/1620<1.9451$. We also extend the anywhere burning result under $L_\infty$ to every fixed dimension $d\ge 3$ to achieve a $\left(2-\frac{1}{2^{d+1}}+\varepsilon\right)$-approximation. Finally, using standard comparisons between planar $L_p$ distances, we transfer our $L_1$ and $L_\infty$ algorithms, together with known Euclidean burning algorithms, to obtain approximation guarantees for every fixed $1\le p\le\infty$.
