关于最小三角形边横截的分布式计算
On Distributed Computation of the Minimum Triangle Edge Transversal
AI总结：
本文研究分布式网络中最小三角形边横截的计算，证明精确求解在CONGEST模型中需近二次方轮数，并提出在poly log n轮内获得(1+ε)-近似及通过归约到超图顶点覆盖实现更快近似的方法。
摘要：
图距离无三角形状态的距离是一个基本的图参数，它计数为了使图变为无三角形而需要从图中移除的边的数量。其对应的计算问题是经典的最小三角形边横截问题，其归一化值是无三角形性质测试算法的基准。虽然无三角形性质测试已经在分布式环境中得到了成功的研究，但据我们所知，在分布式环境中计算该距离本身仍然是未知的，尽管它在集中式环境中已被充分研究。\n本文研究分布式网络中最小三角形边横截的计算。我们通过一个简单的热身构造表明这是一个全局任务，即使在消息不受限制的 LOCAL 模型中也需要 Ω(D) 轮，其中 D 是网络的直径。然而，我们证明对该值进行近似可以快得多。当 n 是网络图的规模时，可以在 poly log n 轮内获得 (1+ε)-近似。此外，通过归约到最小超图顶点覆盖问题，可以获得更快的近似，代价是将近似因子增加到大约 3。在最大度 Δ 的时间开销下，这也可以应用于消息受限的 CONGEST 模型。\n我们的关键技术贡献是证明了在 CONGEST 中计算精确解“难到极致”，需要接近二次方的轮数。由于该问题是一个边选择问题，而之前的下界都是针对节点选择问题的，因此在构造下界时出现了重大挑战，需要我们开发新颖的组成部分。
英文摘要：
The distance of a graph from being triangle-free is a fundamental graph parameter, counting the number of edges that need to be removed from a graph in order for it to become triangle-free. Its corresponding computational problem is the classic minimum triangle edge transversal problem, and its normalized value is the baseline for triangle-freeness testing algorithms. While triangle-freeness testing has been successfully studied in the distributed setting, computing the distance itself in a distributed setting is unknown, to the best of our knowledge, despite being well-studied in the centralized setting. This work addresses the computation of the minimum triangle edge transversal in distributed networks. We show with a simple warm-up construction that this is a global task, requiring $Ω(D)$ rounds even in the $\mathsf{LOCAL}$ model with unbounded messages, where $D$ is the diameter of the network. However, we show that approximating this value can be done much faster. A $(1+ε)$-approximation can be obtained in $\text{poly}\log{n}$ rounds, where $n$ is the size of the network graph. Moreover, faster approximations can be obtained, at the cost of increasing the approximation factor to roughly 3, by a reduction to the minimum hypergraph vertex cover problem. With a time overhead of the maximum degree $Δ$, this can also be applied to the $\mathsf{CONGEST}$ model, in which messages are bounded. Our key technical contribution is proving that computing an exact solution is ``as hard as it gets'' in $\mathsf{CONGEST}$, requiring a near-quadratic number of rounds. Because this problem is an edge selection problem, as opposed to previous lower bounds that were for node selection problems, major challenges arise in constructing the lower bound, requiring us to develop novel ingredients.
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