We study the awake complexity of graph problems that belong to the class O-LOCAL, which includes a large subset of problems solvable by sequential greedy algorithms, such as $(\Delta+1)$-coloring, maximal independent set, maximal matching, etc. 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类图问题的唤醒复杂度，该类包含可通过顺序贪婪算法求解的大量问题，例如$(\Delta+1)$-着色、极大独立集、极大匹配等。已有研究表明，在最大度为$\Delta$的$n$节点图中，O-LOCAL类中的任何问题均可通过确定性分布式算法以$O(\log\Delta+\log^\star n)$的唤醒复杂度求解。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$O(\sqrt{\log n}\cdot\log^\star n)$的唤醒复杂度求解。当$\Delta\gg 2^{\sqrt{\log n}}$时（例如对任意小常数$\epsilon>0$满足$\Delta=n^\epsilon$），该结果实现了相对于现有技术的多项式级改进。达成该结果的关键在于，我们在睡眠模型中构建了使用少量颜色的网络分解，其时间复杂度为亚对数级，这一技术本身具有独立的研究价值。