The connectivity of networks has been widely studied in many high-impact applications, ranging from immunization, critical infrastructure analysis, social network mining, to bioinformatic system studies. Regardless of the end application domains, connectivity minimization has always been a fundamental task to effectively control the functioning of the underlying system. The combinatorial nature of the connectivity minimization problem imposes an exponential computational complexity to find the optimal solution, which is intractable in large systems. To tackle the computational barrier, greedy algorithm is extensively used to ensure a near-optimal solution by exploiting the diminishing returns property of the problem. Despite the empirical success, the theoretical and algorithmic challenges of the problems still remain wide open. On the theoretical side, the intrinsic hardness and the approximability of the general connectivity minimization problem are still unknown except for a few special cases. On the algorithmic side, existing algorithms are hard to balance between the optimization quality and computational efficiency. In this article, we address the two challenges by (1) proving that the general connectivity minimization problem is NP-hard and is the best approximation ratio for any polynomial algorithms, and (2) proposing the algorithm CONTAIN and its variant CONTAIN + that can well balance optimization effectiveness and computational efficiency for eigen-function based connectivity minimization problems in large networks.

中文翻译：

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大规模网络上的快速连接最小化

网络的连通性已在许多高影响力的应用中得到广泛研究，从免疫、关键基础设施分析、社交网络挖掘到生物信息系统研究。无论最终应用领域如何，连接最小化一直是有效控制底层系统功能的一项基本任务。连通性最小化问题的组合性质强加了指数计算复杂性以找到最佳解决方案，这在大型系统中是难以处理的。为了解决计算障碍，贪心算法被广泛用于通过利用问题的收益递减特性来确保接近最优的解决方案。尽管在经验上取得了成功，但这些问题的理论和算法挑战仍然是开放的。在理论上，除了少数特殊情况外，一般连通性最小化问题的内在难度和近似性仍然未知。在算法方面，现有算法很难在优化质量和计算效率之间取得平衡。在本文中，我们通过 (1) 证明一般连通性最小化问题是 NP-hard 和 是任何多项式算法的最佳近似比，并且 (2) 提出算法包含及其变体包含+ 可以很好地平衡大型网络中基于特征函数的连通性最小化问题的优化效果和计算效率。