We study the awake complexity of graph problems that belong to the class O-LOCAL, which includes a subset of problems solvable by sequential greedy algorithms, such as $(\Delta+1)$-coloring and maximal independent set. It is known from previous work that, in $n$-node graphs of maximum degree $\Delta$, any problem in the class O-LOCAL can be solved by a deterministic distributed algorithm with awake complexity $O(\log\Delta+\log^\star n)$. In this paper, we show that any problem belonging to the class O-LOCAL can be solved by a deterministic distributed algorithm with awake complexity $O(\sqrt{\log n}\cdot\log^\star n)$. This leads to a polynomial improvement over the state of the art when $\Delta\gg 2^{\sqrt{\log n}}$, e.g., $\Delta=n^\epsilon$ for some arbitrarily small $\epsilon>0$. The key ingredient for achieving our results is the computation of a network decomposition, that uses a small-enough number of colors, in sub-logarithmic time in the Sleeping model, which can be of independent interest.

翻译：我们研究属于O-LOCAL类图问题的清醒复杂度，该类包含可通过顺序贪婪算法求解的问题子集，例如$(\Delta+1)$-着色和最大独立集。已知在最大度为$\Delta$的$n$节点图中，该类别中的任何问题均可通过确定性分布式算法以$O(\log\Delta+\log^\star n)$的清醒复杂度求解。本文证明，属于O-LOCAL类的任何问题均可通过确定性分布式算法以$O(\sqrt{\log n}\cdot\log^\star n)$的清醒复杂度求解。当$\Delta\gg 2^{\sqrt{\log n}}$时（例如对任意小常数$\epsilon>0$满足$\Delta=n^\epsilon$），该结果相较现有技术实现了多项式级改进。达成该结果的关键在于：在Sleeping模型中，我们以次对数时间计算了使用少量颜色的网络分解，该方法本身具有独立的研究价值。