arXiv 1805.045442018-05-11cs.DS

弦图的分布最小点染色与最大独立集

Distributed Minimum Vertex Coloring and Maximum Independent Set in Chordal Graphs

Christian Konrad, Viktor Zamaraev

摘要：

本文给出了弦图最小顶点着色和弦图最大独立集的确定性分布式$(1+ε)$ -近似算法。我们的着色算法在$O(\frac{1}ε \log n)$轮中运行，并且我们的独立集算法具有$O(\frac{1}ε\log(\frac{1}ε)\log^* n)$轮的运行时间。对于着色，现有的下限意味着对$\frac{1}ε$和$\log n$的依赖性是最好的。对于独立集，我们证明了$O(\frac{1}ε)$轮是必要的.我们的两个算法都利用输入弦图的树分解。它们迭代地剥离区间子图，区间子图通过输入图的树分解来识别，从而将顶点集划分为$O(\log n)$层。对于着色，每个区间图被独立着色，这导致层之间的各种着色冲突。然后，这些冲突在一个单独的阶段中解决，使用我们划分的特定结构。对于独立集，仅需要第一$O( \log \frac{1}ε)$层，因为它们已经包含足够大的独立集。我们开发了一个$(1+ε)$近似最大独立集算法的区间图，然后我们应用到这些层。这项工作提出了一个问题，如何有用的树分解是分布式计算。

英文摘要：

We give deterministic distributed $(1+ε)$-approximation algorithms for Minimum Vertex Coloring and Maximum Independent Set on chordal graphs in the LOCAL model. Our coloring algorithm runs in $O(\frac{1}ε \log n)$ rounds, and our independent set algorithm has a runtime of $O(\frac{1}ε\log(\frac{1}ε)\log^* n)$ rounds. For coloring, existing lower bounds imply that the dependencies on $\frac{1}ε$ and $\log n$ are best possible. For independent set, we prove that $O(\frac{1}ε)$ rounds are necessary. Both our algorithms make use of a tree decomposition of the input chordal graph. They iteratively peel off interval subgraphs, which are identified via the tree decomposition of the input graph, thereby partitioning the vertex set into $O(\log n)$ layers. For coloring, each interval graph is colored independently, which results in various coloring conflicts between the layers. These conflicts are then resolved in a separate phase, using the particular structure of our partitioning. For independent set, only the first $O( \log \frac{1}ε)$ layers are required as they already contain a large enough independent set. We develop a $(1+ε)$-approximation maximum independent set algorithm for interval graphs, which we then apply to those layers. This work raises the question as to how useful tree decompositions are for distributed computing.

机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。

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英文摘要：

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