By far the most fruitful technique for showing lower bounds for the CONGEST model is reductions to two-party communication complexity. This technique has yielded nearly tight results for various fundamental problems such as distance computations, minimum spanning tree, minimum vertex cover, and more. In this work, we take this technique a step further, and we introduce a framework of reductions to $t$-party communication complexity, for every $t\geq 2$. Our framework enables us to show improved hardness results for maximum independent set. Recently, Bachrach et al.[PODC 2019] used the two-party framework to show hardness of approximation for maximum independent set. They show that finding a $(5/6+\epsilon)$-approximation requires $\Omega(n/\log^6 n)$ rounds, and finding a $(7/8+\epsilon)$-approximation requires $\Omega(n^2/\log^7 n)$ rounds, in the CONGEST model where $n$ in the number of nodes in the network. We improve the results of Bachrach et al. by using reductions to multi-party communication complexity. Our results: (1) Any algorithm that finds a $(1/2+\epsilon)$-approximation for maximum independent set in the CONGEST model requires $\Omega(n/\log^3 n)$ rounds. (2) Any algorithm that finds a $(3/4+\epsilon)$-approximation for maximum independent set in the CONGEST model requires $\Omega(n^2/\log^3 n)$ rounds.

中文翻译：

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超越 Alice 和 Bob：改进了 CONGEST 中最大独立集的不逼近性

到目前为止，显示 CONGEST 模型下界最有效的技术是降低两方通信的复杂性。对于距离计算、最小生成树、最小顶点覆盖等各种基本问题，这种技术已经产生了近乎严格的结果。在这项工作中，我们将这项技术更进一步，我们引入了一个减少 $t$-party 通信复杂性的框架，对于每个 $t\geq 2$。我们的框架使我们能够显示最大独立集的改进硬度结果。最近，Bachrach 等人 [PODC 2019] 使用两方框架来展示最大独立集的近似难度。他们表明，找到 $(5/6+\epsilon)$-近似值需要 $\Omega(n/\log^6 n)$ 轮，而找到 $(7/8+\epsilon)$-近似值需要 $ \Omega(n^2/\log^7 n)$ 轮，在 CONGEST 模型中，网络中的节点数为 $n$。我们改进了 Bachrach 等人的结果。通过降低多方通信的复杂性。我们的结果：（1）任何在 CONGEST 模型中找到最大独立集的 $(1/2+\epsilon)$-近似值的算法都需要 $\Omega(n/\log^3 n)$ 轮。(2) 任何在 CONGEST 模型中找到最大独立集的 $(3/4+\epsilon)$-近似值的算法都需要 $\Omega(n^2/\log^3 n)$ 轮。