The VertexCover problem is proven to be computationally hard in different ways: It is NP-complete to find an optimal solution and even NP-hard to find an approximation with reasonable factors. In contrast, recent experiments suggest that on many real-world networks the run time to solve VertexCover is way smaller than even the best known FPT-approaches can explain. Similarly, greedy algorithms deliver very good approximations to the optimal solution in practice. We link these observations to two properties that are observed in many real-world networks, namely a heterogeneous degree distribution and high clustering. To formalize these properties and explain the observed behavior, we analyze how a branch-and-reduce algorithm performs on hyperbolic random graphs, which have become increasingly popular for modeling real-world networks. In fact, we are able to show that the VertexCover problem on hyperbolic random graphs can be solved in polynomial time, with high probability. The proof relies on interesting structural properties of hyperbolic random graphs. Since these predictions of the model are interesting in their own right, we conducted experiments on real-world networks showing that these properties are also observed in practice. When utilizing the same structural properties in an adaptive greedy algorithm, further experiments suggest that, on real instances, this leads to better approximations than the standard greedy approach within reasonable time.

中文翻译：

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在多项式时间内求解双曲随机图上的顶点覆盖

VertexCover 问题在不同方面被证明在计算上是困难的：找到最优解是 NP 完全的，甚至是 NP 难找到具有合理因子的近似值。相比之下，最近的实验表明，在许多现实世界的网络上，解决 VertexCover 的运行时间甚至比最知名的 FPT 方法所能解释的要小得多。类似地，贪婪算法在实践中为最优解提供了非常好的近似。我们将这些观察结果与在许多现实世界网络中观察到的两个属性联系起来，即异构度分布和高聚类。为了形式化这些属性并解释观察到的行为，我们分析了分支归约算法如何在双曲随机图上执行，双曲随机图在模拟现实世界网络中变得越来越流行。实际上，我们能够证明双曲随机图上的 VertexCover 问题可以在多项式时间内以高概率解决。证明依赖于双曲随机图的有趣结构特性。由于模型的这些预测本身就很有趣，因此我们在现实世界的网络上进行了实验，表明在实践中也观察到了这些特性。当在自适应贪婪算法中使用相同的结构特性时，进一步的实验表明，在实际情况下，这会在合理的时间内比标准贪婪方法产生更好的近似值。由于模型的这些预测本身就很有趣，因此我们在现实世界的网络上进行了实验，表明在实践中也观察到了这些特性。当在自适应贪婪算法中使用相同的结构特性时，进一步的实验表明，在实际情况下，这会在合理的时间内比标准贪婪方法产生更好的近似值。由于模型的这些预测本身就很有趣，因此我们在现实世界的网络上进行了实验，表明在实践中也观察到了这些特性。当在自适应贪婪算法中使用相同的结构特性时，进一步的实验表明，在实际情况下，这会在合理的时间内比标准贪婪方法产生更好的近似值。