平面图中近最优分布式支配集

中文摘要 AI 辅助

我们给出一个确定性算法，在LOCAL模型的常数轮内，对平面图上的最小支配集问题实现$(8+\varepsilon)$-近似，适用于任意$\varepsilon>0$。这改进了Heydt等人先前得到的$11+\varepsilon$的比率。该比率在此模型中接近最优：其前导常数仅比已知下界$7$高1。我们的结果将先前差距的四分之三缩小，将其从$4$降至$1$。我们的主要贡献是一个锐利的结构界。对于任意支配集$D$，将$D$外的每个顶点分配给一个相邻的中心，得到不相交的拥有者块。若$k_x$统计包含$x$的邻居的其他块的数量，则$\sum_{x\notin D}(k_x-2)^+\le(4|D|-12)^+$，其中$z^+=\max\{z,0\}$。该界对任意这样的分配成立，且对任意大的最小支配集等号成立。我们将此界用于他们的三阶段框架，采用新参数和相同的最终线性规划过程。该算法既不需要平面嵌入，也不需要图的大小，其轮数界仅依赖于$\varepsilon$。Bonamy等人的传递定理还给出了在有界欧拉亏格图上确定性$(25+\varepsilon)$-近似，轮数界仅依赖于$\varepsilon$和亏格。

英文摘要

We give a deterministic $(8+\varepsilon)$-approximation for minimum dominating set on planar graphs in a constant number of rounds of the LOCAL model, for every $\varepsilon>0$. This improves the previous ratio $11+\varepsilon$ obtained by Heydt et al. The ratio is near-optimal in this model: its leading constant is only one above the known lower bound of $7$. Our result closes three quarters of the previous gap, reducing it from $4$ to $1$. Our main contribution is a sharp structural bound. For any dominating set $D$, assigning each vertex outside $D$ to a neighboring center gives disjoint owner blocks. If $k_x$ counts the other blocks containing a neighbor of $x$, then $\sum_{x\notin D}(k_x-2)^+\le(4|D|-12)^+$, where $z^+=\max\{z,0\}$. The bound holds for every such assignment, and equality holds for arbitrarily large minimum dominating sets. We use this bound in their three-phase framework, with new parameters and the same final linear-programming procedure. The algorithm requires neither a planar embedding nor the graph size, and its round bound depends only on $\varepsilon$. The transfer theorem of Bonamy et al. also gives a deterministic $(25+\varepsilon)$-approximation on graphs of bounded Euler genus, with a round bound depending only on $\varepsilon$ and the genus.

发表机构

• Adam Mickiewicz University（亚当·密茨凯维奇大学）

机构由 AI 辅助整理，请以论文原文为准。

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19 pages, 7 figures