有界扩张图类的分布控制
Distributed Domination on Graph Classes of Bounded Expansion
摘要：
对于有界扩张图类上的（连通）距离- $r$控制集问题，给出了一个新的常数因子逼近算法。有界扩展类包括许多常见的稀疏图类，如平面图和排除（拓扑）子图，值得注意的是，这些类形成最一般的子图封闭类的图形，其中距离- $r$支配集问题的顺序常数因子近似算法是目前已知的。我们的算法可以在分布式计算的\congestbc模型中实现，并使用$\mathcal{O}(r^2 \log n)$通信轮。我们的技术，这可能是独立的利益，是基于一个分布式计算的稀疏邻域覆盖的小半径有界扩展类。本文给出了如何计算某个函数$f$在$\mathcal{O}(r^2 \log n)$通信轮中的每类有界展开式上的半径$r$ -邻域覆盖和重叠$f(r)$.在$\mathcal{CONGEST}_{\mathrm{BC}}$模型中。最后，我们展示了如何使用更大的权力的$\mathcal{LOCAL}$模型，把任何距离$r$控制集到一个不断更大的连接距离$r$控制集在$3r+1$轮上的任何类的有界扩展。结合该算法，例如，利用Lenzen等人关于平面图控制集的常数因子近似算法给出了$\mathcal{LOCAL}$模型中平面图的连通控制集的常圈数因子近似算法，其近似比仅为Lenzen等人的近似比的$6$倍.的算法。
英文摘要：
We provide a new constant factor approximation algorithm for the (connected) distance-$r$ dominating set problem on graph classes of bounded expansion. Classes of bounded expansion include many familiar classes of sparse graphs such as planar graphs and graphs with excluded (topological) minors, and notably, these classes form the most general subgraph closed classes of graphs for which a sequential constant factor approximation algorithm for the distance-$r$ dominating set problem is currently known. Our algorithm can be implemented in the \congestbc model of distributed computing and uses $\mathcal{O}(r^2 \log n)$ communication rounds. Our techniques, which may be of independent interest, are based on a distributed computation of sparse neighborhood covers of small radius on bounded expansion classes. We show how to compute an $r$-neighborhood cover of radius~$2r$ and overlap $f(r)$ on every class of bounded expansion in $\mathcal{O}(r^2 \log n)$ communication rounds for some function~$f$.% in the $\mathcal{CONGEST}_{\mathrm{BC}}$ model. Finally, we show how to use the greater power of the $\mathcal{LOCAL}$ model to turn any distance-$r$ dominating set into a constantly larger connected distance-$r$ dominating set in $3r+1$ rounds on any class of bounded expansion. Combining this algorithm, e.g., with the constant factor approximation algorithm for dominating sets on planar graphs of Lenzen et al.\ gives a constant factor approximation algorithm for connected dominating sets on planar graphs in a constant number of rounds in the $\mathcal{LOCAL}$ model, where the approximation ratio is only $6$ times larger than that of Lenzen et al.'s algorithm.
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