AI 中文总结
本研究在睡眠CONGEST模型中，针对最小支配集问题提出了唤醒复杂度更低的近似算法，还推广了集合覆盖算法并结合采样、虚拟二叉树调度等技术，实现了唤醒复杂度与近似比的权衡。
AI 中文摘要
我们在睡眠CONGEST模型（Chatterjee、Gmyr和Pandurangan，PODC 2020）中研究最小支配集（MDS）问题，该模型是标准CONGEST模型的推广，其中节点可在部分轮次中处于睡眠状态，仅在唤醒时才能执行计算、发送或接收消息操作。算法在该模型中的唤醒复杂度是指，在算法执行过程中，节点被唤醒的轮次的最坏情况数量（针对所有输入和所有节点）。虽然现有多种针对MDS的O(log Δ)近似算法（期望意义下）可在O(log² Δ)轮次内运行，但所有这些算法的唤醒复杂度均为Ω(log² Δ)。能否降低该唤醒复杂度是本研究的核心问题。我们提出了首个针对MDS的O(log Δ)近似算法，其唤醒复杂度为o(log² Δ)；该算法在O(log² Δ)轮次内运行，唤醒复杂度为Õ(log Δ)。我们还可进一步降低唤醒复杂度，但需以近似比为代价：针对任意1<α≤Δ，我们在睡眠CONGEST模型中提出一种算法，其可在Õ(log Δ·log_α Δ)轮次内、以Õ(log_α Δ)的唤醒复杂度，计算出期望意义下的O(α log Δ)近似支配集。我们的研究基于对Grunau、Mitrović、Rubinfeld和Vakilian（SODA 2020）的CONGEST模型集合覆盖算法的推广，该推广针对参数1<p,q≤Δ，可在O(log_p Δ·log_q Δ)轮次内计算出O(p·q·log_p Δ)近似支配集。我们的睡眠CONGEST算法将多种技术（包括基于采样的估计、使用虚拟二叉树的调度）应用于上述双参数集合覆盖算法。
英文摘要
We study the Minimum Dominating Set (MDS) problem in the sleeping CONGEST model (Chatterjee, Gmyr, and Pandurangan, PODC 2020), a generalization of the standard CONGEST model, in which a node may sleep in some rounds and can only compute, send messages, or receive messages when it is awake. The awake complexity of an algorithm in this model is the worst case number (over all inputs and all nodes) of rounds a node is awake for during the execution of the algorithm. While there are several $O(\log Δ)$-approximation algorithms (in expectation) for MDS that run in $O(\log^2 Δ)$ rounds, all of these have $Ω(\log^2 Δ)$ awake complexity. Whether this awake complexity can be improved is the question that drives our work. We present the first $O(\log Δ)$-approximation algorithm for MDS with $o(\log^2 Δ)$ awake complexity; our algorithm runs in $O(\log^2Δ)$ rounds with $\tilde{O}(\logΔ)$ awake complexity. We can reduce the awake complexity further, but at the cost of approximation: we present, for any $1<α\leΔ$, an algorithm in the sleeping CONGEST model that computes an $O(α\logΔ)$-approximate dominating set in expectation in $\tilde{O}(\logΔ\cdot \log_α Δ)$ rounds with $\tilde{O}(\log_α Δ)$ awake complexity. Our results depend on a generalization of the CONGEST model SetCover algorithm of Grunau, Mitrovi'c, Rubinfeld, and Vakilian (SODA 2020) that we develop. This generalization computes an $O(p\cdot q\cdot\log_pΔ)$-approximate dominating set in $O(\log_pΔ\cdot\log_qΔ)$ rounds for parameters $1<p,q\leΔ$. Our sleeping CONGEST algorithms apply a variety of techniques including sampling-based estimation and scheduling using virtual binary trees to the aforementioned 2-parameter SetCover algorithm.
CommentsFull version of a paper to appear in the 40th International Symposium on Distributed Computing (DISC 2026)
