有界荫度图中最小支配集问题的简单、快速和组合算法
Algorithms for the Minimum Dominating Set Problem in Bounded Arboricity Graphs: Simpler, Faster, and Combinatorial
摘要：
本文重新讨论荫度为$α$的图的最小支配集问题。Bansal和Umboh [BU 17]给出了一个$O(α)$ -近似LP舍入算法，该算法也可以转换为一个近线性时间算法，使用显式混合填充和覆盖或纯覆盖LP的通用近似结果[KY 14，You 14，AZO 19，JP 10]。此外，[BU 17]表明，实现近似因子的渐近改进是NP困难的。另一方面，Lenzen和Wattenhofer [LW 10]和Jones等人[JLR+13]的前两种非基于LP的算法在线性时间内实现了$O(α^2)$的近似因子。在分布式设置中存在类似的情况：虽然[KMW 06]中隐含了$O(\log^2 n)$轮基于LP的$O(α)$近似算法，但Lenzen和Wattenhofer [LW 10]的最佳非基于LP的算法是其集中式算法的实现，在$O(\log n)$轮内提供$O(α^2)$近似。我们解决的问题，是否可以实现一个$O(α)$ -近似算法，这不是LP为基础的，无论是在集中式设置或在分布式设置。我们肯定地解决了这两个问题，并在途中实现了比最先进的基于LP的算法更快的算法。更具体地说，我们的贡献是双重的：1。在集中式设置中，我们提供了一个令人惊讶的简单的组合算法，是渐近最佳的近似因子和运行时间：一个$O(α)$近似线性时间。2.基于我们的集中式算法，我们设计了一个分布式的组合$O(α)$ -近似算法在CONGEST模型，运行在$O(α\log n )$轮与高概率。
英文摘要：
We revisit the minimum dominating set problem on graphs with arboricity bounded by $α$. Bansal and Umboh [BU17] gave an $O(α)$-approximation LP rounding algorithm, which also translates into a near-linear time algorithm using general-purpose approximation results for explicit mixed packing and covering or pure covering LPs [KY14, You14, AZO19, Qua10]. Moreover, [BU17] showed that it is NP-hard to achieve an asymptotic improvement for the approximation factor. On the other hand, the previous two non-LP-based algorithms, by Lenzen and Wattenhofer [LW10], and Jones et al. [JLR+13], achieve an approximation factor of $O(α^2)$ in linear time. There is a similar situation in the distributed setting: While there is an $O(\log^2 n)$-round LP-based $O(α)$-approximation algorithm implied in [KMW06], the best non-LP-based algorithm by Lenzen and Wattenhofer [LW10] is an implementation of their centralized algorithm, providing an $O(α^2)$-approximation within $O(\log n)$ rounds. We address the questions of whether one can achieve an $O(α)$-approximation algorithm that is not LP-based, either in the centralized setting or in the distributed setting. We resolve both questions in the affirmative, and en route achieve algorithms that are faster than the state-of-the-art LP-based algorithms. More specifically, our contribution is two-fold: 1. In the centralized setting, we provide a surprisingly simple combinatorial algorithm that is asymptotically optimal in terms of both approximation factor and running time: an $O(α)$-approximation in linear time. 2. Based on our centralized algorithm, we design a distributed combinatorial $O(α)$-approximation algorithm in the CONGEST model that runs in $O(α\log n )$ rounds with high probability.
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