In this paper, we present a low-diameter decomposition algorithm in the LOCAL model of distributed computing that succeeds with probability $1 - 1/poly(n)$. Specifically, we show how to compute an $\left(\epsilon, O\left(\frac{\log n}{\epsilon}\right)\right)$ low-diameter decomposition in $O\left(\frac{\log^3(1/\epsilon)\log n}{\epsilon}\right)$ round Further developing our techniques, we show new distributed algorithms for approximating general packing and covering integer linear programs in the LOCAL model. For packing problems, our algorithm finds an $(1-\epsilon)$-approximate solution in $O\left(\frac{\log^3 (1/\epsilon) \log n}{\epsilon}\right)$ rounds with probability $1 - 1/poly(n)$. For covering problems, our algorithm finds an $(1+\epsilon)$-approximate solution in $O\left(\frac{\left(\log \log n + \log (1/\epsilon)\right)^3 \log n}{\epsilon}\right)$ rounds with probability $1 - 1/poly(n)$. These results improve upon the previous $O\left(\frac{\log^3 n}{\epsilon}\right)$-round algorithm by Ghaffari, Kuhn, and Maus [STOC 2017] which is based on network decompositions. Our algorithms are near-optimal for many fundamental combinatorial graph optimization problems in the LOCAL model, such as minimum vertex cover and minimum dominating set, as their $(1\pm \epsilon)$-approximate solutions require $\Omega\left(\frac{\log n}{\epsilon}\right)$ rounds to compute.

翻译：本文提出了一种在分布式计算LOCAL模型中的低直径分解算法，其成功概率为$1 - 1/poly(n)$。具体而言，我们展示了如何在$O\left(\frac{\log^3(1/\epsilon)\log n}{\epsilon}\right)$轮内计算出一个$\left(\epsilon, O\left(\frac{\log n}{\epsilon}\right)\right)$低直径分解。进一步改进技术后，我们提出了在LOCAL模型中近似一般包装与覆盖整数线性规划的分布式新算法。对于包装问题，算法在$O\left(\frac{\log^3 (1/\epsilon) \log n}{\epsilon}\right)$轮内以概率$1 - 1/poly(n)$找到$(1-\epsilon)$近似解。对于覆盖问题，算法在$O\left(\frac{\left(\log \log n + \log (1/\epsilon)\right)^3 \log n}{\epsilon}\right)$轮内以概率$1 - 1/poly(n)$找到$(1+\epsilon)$近似解。这些结果改进了Ghaffari、Kuhn和Maus [STOC 2017]基于网络分解的先前$O\left(\frac{\log^3 n}{\epsilon}\right)$轮算法。对于LOCAL模型中的许多基本组合图优化问题（如最小顶点覆盖和最小支配集），我们的算法近乎最优，因为其$(1\pm \epsilon)$近似解需要$\Omega\left(\frac{\log n}{\epsilon}\right)$轮计算。