Finding a minimum vertex cover in a network is a fundamental NP-complete graph problem. One way to deal with its computational hardness, is to trade the qualitative performance of an algorithm (allowing non-optimal outputs) for an improved running time. For the vertex cover problem, there is a gap between theory and practice when it comes to understanding this tradeoff. On the one hand, it is known that it is NP-hard to approximate a minimum vertex cover within a factor of $\sqrt{2}$. On the other hand, a simple greedy algorithm yields close to optimal approximations in practice. A promising approach towards understanding this discrepancy is to recognize the differences between theoretical worst-case instances and real-world networks. Following this direction, we close the gap between theory and practice by providing an algorithm that efficiently computes nearly optimal vertex cover approximations on hyperbolic random graphs; a network model that closely resembles real-world networks in terms of degree distribution, clustering, and the small-world property. More precisely, our algorithm computes a $(1 + o(1))$-approximation, asymptotically almost surely, and has a running time of $\mathcal{O}(m \log(n))$. The proposed algorithm is an adaptation of the successful greedy approach, enhanced with a procedure that improves on parts of the graph where greedy is not optimal. This makes it possible to introduce a parameter that can be used to tune the tradeoff between approximation performance and running time. Our empirical evaluation on real-world networks shows that this allows for improving over the near-optimal results of the greedy approach.

翻译：摘要：寻找网络中的最小顶点覆盖是一个基础性的NP完全图问题。处理其计算难解性的一种方式，是牺牲算法的定性性能（允许非最优输出）以换取更优的运行时间。对于顶点覆盖问题，理解这种权衡时存在理论与实践之间的鸿沟。一方面，已知在$\sqrt{2}$因子内近似最小顶点覆盖是NP难的。另一方面，简单的贪心算法在实践中能产生接近最优的近似解。弥合这一差异的一个有前景的方法是，认识到理论最坏情形实例与现实网络之间的区别。遵循这一方向，我们通过提出一种算法来缩小理论与实践之间的差距，该算法能在双曲随机图上高效计算近乎最优的顶点覆盖近似——这一网络模型在度分布、聚类系数和小世界特性上高度接近真实网络。具体而言，我们的算法可渐进几乎必然地计算$(1 + o(1))$-近似，运行时间复杂度为$\mathcal{O}(m \log(n))$。该算法是对成功贪心方法的改进，并增强了一个过程来优化贪心策略非最优的图区域。这使得引入一个可调参数成为可能，用于权衡近似性能与运行时间。在真实网络上的实证评估表明，该算法能够进一步改善贪心算法已接近最优的结果。