Large real-world networks typically follow a power-law degree distribution. To study such networks, numerous random graph models have been proposed. However, real-world networks are not drawn at random. Therefore, Brach et al. (27th symposium on discrete algorithms (SODA), pp 1306–1325, 2016) introduced two natural deterministic conditions: (1) a power-law upper bound on the degree distribution (PLB-U) and (2) power-law neighborhoods, that is, the degree distribution of neighbors of each vertex is also upper bounded by a power law (PLB-N). They showed that many real-world networks satisfy both properties and exploit them to design faster algorithms for a number of classical graph problems. We complement their work by showing that some well-studied random graph models exhibit both of the mentioned PLB properties. PLB-U and PLB-N hold with high probability for Chung–Lu Random Graphs and Geometric Inhomogeneous Random Graphs and almost surely for Hyperbolic Random Graphs. As a consequence, all results of Brach et al. also hold with high probability or almost surely for those random graph classes. In the second part we study three classical $$\textsf {NP}$$ NP -hard optimization problems on PLB networks. It is known that on general graphs with maximum degree $$\Delta$$ Δ , a greedy algorithm, which chooses nodes in the order of their degree, only achieves a $$\Omega (\ln \Delta )$$ Ω ( ln Δ ) -approximation for Minimum Vertex Cover and Minimum Dominating Set , and a $$\Omega (\Delta )$$ Ω ( Δ ) -approximation for Maximum Independent Set . We prove that the PLB-U property with $$\beta >2$$ β > 2 suffices for the greedy approach to achieve a constant-factor approximation for all three problems. We also show that these problems are APX -hard even if PLB-U, PLB-N, and an additional power-law lower bound on the degree distribution hold. Hence, a PTAS cannot be expected unless P = NP . Furthermore, we prove that all three problems are in MAX SNP if the PLB-U property holds.

中文翻译：

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贪婪有利于确定性无标度网络

大型现实世界网络通常遵循幂律度分布。为了研究这样的网络，已经提出了许多随机图模型。然而，现实世界的网络并不是随机绘制的。因此，布拉赫等人。（第 27 届离散算法研讨会 (SODA)，第 1306–1325 页，2016 年）介绍了两个自然确定性条件：(1) 度分布 (PLB-U) 的幂律上限和 (2) 幂律邻域，也就是说，每个顶点的邻居的度分布也有一个幂律（PLB-N）的上限。他们表明，许多现实世界的网络都满足这两个属性，并利用它们为许多经典图问题设计更快的算法。我们通过展示一些经过充分研究的随机图模型展示了提到的 PLB 属性来补充他们的工作。PLB-U 和 PLB-N 对 Chung-Lu 随机图和几何非齐次随机图的概率很高，对于双曲随机图几乎可以肯定。因此，Brach 等人的所有结果。对于那些随机图类，也很有可能或几乎可以肯定地成立。在第二部分中，我们研究了 PLB 网络上的三个经典 $$\textsf {NP}$$ NP -hard 优化问题。已知在最大度数为 $$\Delta$$ Δ 的一般图上，按照度数顺序选择节点的贪心算法只实现了 $$\Omega (\ln \Delta )$$ Ω ( ln Δ ) - 最小顶点覆盖和最小支配集的近似，以及最大独立集的 $$\Omega (\Delta )$$ Ω (Δ) -近似。我们证明了 $$\beta >2$$ β > 2 足以让贪婪方法实现所有三个问题的常数因子近似。我们还表明，即使 PLB-U、PLB-N 和度分布的附加幂律下界成立，这些问题也是 APX 难的。因此，除非 P = NP，否则不能期待 PTAS。此外，如果 PLB-U 属性成立，我们证明所有三个问题都在 MAX SNP 中。