我们描述了一个简单的确定性\(O( \varepsilon ^{-1} \log \Delta )\)轮分布式算法,用于\((2\alpha +1)(1 + \varepsilon )\)最小加权支配集的近似在最多具有树木性的图上\(\alpha \)。这里\(\Delta \)表示最大度数。我们还展示了一个下界,证明即使对于未加权的情况,该轮复杂度也几乎是最优的,通过减少分布式顶点覆盖近似上著名的 KMW 下界(Kuhn 等人,JACM 63:116, 2016)。我们的算法改进了之前的所有结果(仅适用于未加权图),包括在\(O(\log n) \) 轮中的随机 \(O(\alpha ^2) \)近似(Lenzen 等人在 International分布式计算研讨会,Springer,2010),\(O(\log \Delta )\) 轮中的确定性 \ (O(\alpha \log \Delta )\)近似(Lenzen 等人在分布式计算国际研讨会上, Springer, 2010), \ (O(\log ^2 \Delta )\) 轮中的确定性\(O(\alpha )\)近似(隐含在 Bansal 等人的 Inform Process Lett 122:21–24 中, 2017 年第 17 届离散算法研讨会 (SODA),2006),以及\(O(\alpha \log n )\) 轮中的随机\(O(\alpha )\)近似(Morgan 等人,第 35 届国际会议)分布式计算研讨会,2021)。我们还提供了一种随机\(O(\alpha \log \Delta )\)轮分布式算法,将近似因子锐化为\(\alpha (1+o(1))\)。如果每个节点被限制进行多项式时间计算,我们的近似因子在一阶上是紧的,因为它是 NP-hard 实现\(\alpha - 1 - \varepsilon \)近似(Bansal 等人在 Inform Process Lett 中) 122:21-24, 2017)。
"点击查看英文标题和摘要"
Near-optimal distributed dominating set in bounded arboricity graphs
We describe a simple deterministic \(O( \varepsilon ^{-1} \log \Delta )\) round distributed algorithm for \((2\alpha +1)(1 + \varepsilon )\) approximation of minimum weighted dominating set on graphs with arboricity at most \(\alpha \). Here \(\Delta \) denotes the maximum degree. We also show a lower bound proving that this round complexity is nearly optimal even for the unweighted case, via a reduction from the celebrated KMW lower bound on distributed vertex cover approximation (Kuhn et al. in JACM 63:116, 2016). Our algorithm improves on all the previous results (that work only for unweighted graphs) including a randomized \(O(\alpha ^2)\) approximation in \(O(\log n)\) rounds (Lenzen et al. in International symposium on distributed computing, Springer, 2010), a deterministic \(O(\alpha \log \Delta )\) approximation in \(O(\log \Delta )\) rounds (Lenzen et al. in international symposium on distributed computing, Springer, 2010), a deterministic \(O(\alpha )\) approximation in \(O(\log ^2 \Delta )\) rounds (implicit in Bansal et al. in Inform Process Lett 122:21–24, 2017; Proceeding 17th symposium on discrete algorithms (SODA), 2006), and a randomized \(O(\alpha )\) approximation in \(O(\alpha \log n)\) rounds (Morgan et al. in 35th International symposiumon distributed computing, 2021). We also provide a randomized \(O(\alpha \log \Delta )\) round distributed algorithm that sharpens the approximation factor to \(\alpha (1+o(1))\). If each node is restricted to do polynomial-time computations, our approximation factor is tight in the first order as it is NP-hard to achieve \(\alpha - 1 - \varepsilon \) approximation (Bansal et al. in Inform Process Lett 122:21-24, 2017).