We design distributed algorithms to compute approximate solutions for several related graph optimization problems. All our algorithms have round complexity being logarithmic in the number of nodes of the underlying graph and in particular independent of the graph diameter. By using a primal–dual approach, we develop a $2(1+ϵ)$-approximation algorithm for computing the coreness values of the nodes in the underlying graph, as well as a $2(1+ϵ)$-approximation algorithm for the min–max edge orientation problem, where the goal is to orient the edges so as to minimize the maximum weighted in-degree. We provide lower bounds showing that the aforementioned algorithms are tight both in terms of the approximation guarantee and the round complexity. Additionally, motivated by the fact that the densest subset problem has an inherent dependency on the diameter of the graph, we study a weaker version that does not suffer from the same limitation. Finally, we conduct experiments on large real-world graphs to evaluate the effectiveness of our algorithms.

我们设计了分布式算法来为几个相关的图优化问题计算近似解。我们所有算法的舍入复杂度在基础图的节点数上都是对数的，尤其是与图的直径无关。通过使用原始对偶方法，我们开发了$2（\mathrm{1个}+ϵ）$-近似算法，用于计算基础图中节点的核心值，以及 $2（\mathrm{1个}+ϵ）$-近似算法用于最小-最大边缘定向问题，该目标的目的是使边缘定向，以便最大程度减小最大加权入度。我们提供了下界，表明上述算法在逼近保证和舍入复杂度方面都比较严格。此外，受最密集子集问题与图的直径有内在依赖关系这一事实的启发，我们研究了没有相同限制的较弱版本。最后，我们在大型现实世界图上进行实验，以评估算法的有效性。