SIAM Journal on Computing, Volume 50, Issue 3, Page 1103-1147, January 2021.
We address the fundamental network design problem of constructing approximate minimum spanners. Our contributions are for the distributed setting, providing both algorithmic and hardness results. Our main hardness result shows that an $\alpha$-approximation for the minimum directed $k$-spanner problem for $k \geq 5$ requires $\Omega(n /\sqrt{\alpha}\log{n})$ rounds using deterministic algorithms or $\Omega(\sqrt{n }/\sqrt{\alpha}\log{n})$ rounds using randomized ones, in the Congest model of distributed computing. Combined with the constant-round $O(n^{\epsilon})$-approximation algorithm in the Local model of [L. Barenboim, M. Elkin, and C. Gavoille, Theoret. Comput. Sci., 751 (2016), pp. 2--23], as well as a polylog-round $(1+\epsilon)$-approximation algorithm in the Local model that we show here, our lower bounds for the Congest model imply a strict separation between the Local and Congest models. Notably, to the best of our knowledge, this is the first separation between these models for a local approximation problem. Similarly, a separation between the directed and undirected cases is implied. We also prove hardness results for weighted $k$-spanners and for unweighted undirected $k$-spanners for $k \geq 4$ in the Congest model. In addition, we show lower bounds for the minimum weighted 2-spanner problem in the Congest and Local models. On the algorithmic side, apart from the aforementioned $(1+\epsilon)$-approximation algorithm for minimum $k$-spanners, our main contribution is a new distributed construction of minimum 2-spanners that uses only polynomial local computations. Our algorithm has a guaranteed approximation ratio of $O(\log(m/n))$ for a graph with $n$ vertices and $m$ edges, which matches the best known ratio for polynomial-time sequential algorithms [G. Kortsarz and D. Peleg, J. Algorithms, 17 (1994), pp. 222--236], and is tight if we restrict ourselves to polynomial local computations. An algorithm with this approximation factor was not previously known for the distributed setting. The number of rounds required for our algorithm is $O(\log{n}\log{\Delta})$ with high probability, where $\Delta$ is the maximum degree in the graph. Our approach allows us to extend our algorithm to work also for the directed, weighted, and client-server variants of the problem. It also provides a Congest algorithm for the minimum dominating set problem, with a guaranteed $O(\log{\Delta})$ approximation ratio.

中文翻译：

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分布式扳手近似

SIAM Journal on Computing，第 50 卷，第 3 期，第 1103-1147 页，2021 年 1 月。
我们解决了构建近似最小扳手的基本网络设计问题。我们的贡献是针对分布式设置，提供算法和硬度结果。我们的主要硬度结果表明，$k \geq 5$ 的最小有向 $k$-spanner 问题的 $\alpha$-近似需要 $\Omega(n /\sqrt{\alpha}\log{n})$在分布式计算的拥塞模型中，使用确定性算法的轮次或使用随机算法的 $\Omega(\sqrt{n }/\sqrt{\alpha}\log{n})$ 轮次。结合 [L. Barenboim、M. Elkin 和 C. Gavoille，理论。计算。Sci., 751 (2016), pp. 2--23]，以及我们在此处展示的本地模型中的多对数轮 $(1+\epsilon)$-近似算法，我们对 Congest 模型的下限意味着 Local 和 Congest 模型之间的严格分离。值得注意的是，据我们所知，这是针对局部逼近问题的这些模型之间的第一次分离。类似地，暗示了有向和无向事例之间的分离。我们还证明了 Congest 模型中 $k\geq 4$ 的加权 $k$-spanners 和未加权无向 $k$-spanners 的硬度结果。此外，我们在 Congest 和 Local 模型中显示了最小加权 2-spanner 问题的下限。在算法方面，除了前面提到的用于最小 $k$-spanner 的 $(1+\epsilon)$-近似算法之外，我们的主要贡献是一种新的最小 2-spanner 分布式构造，它仅使用多项式局部计算。对于具有 $n$ 个顶点和 $m$ 个边的图，我们的算法保证了 $O(\log(m/n))$ 的近似比率，这与多项式时间序列算法的最佳已知比率相匹配 [G. Kortsarz 和 D. Peleg, J. Algorithms, 17 (1994), pp. 222--236]，如果我们将自己限制在多项式局部计算中，则它是严格的。具有此近似因子的算法以前并不适用于分布式设置。我们的算法所需的轮数为 $O(\log{n}\log{\Delta})$ 的概率很高，其中 $\Delta$ 是图中的最大度数。我们的方法允许我们扩展我们的算法，使其也适用于问题的定向、加权和客户端-服务器变体。它还为最小支配集问题提供了一个拥塞算法，具有保证的 $O(\log{\Delta})$ 近似比。