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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.

Network connectivity optimization, which aims to manipulate network connectivity by changing its underlying topology, is a fundamental task behind a wealth of high-impact data mining applications, ranging from immunization, critical infrastructure construction, social collaboration mining, bioinformatics analysis, to intelligent transportation system design. To tackle its exponential computation complexity, greedy algorithms have been extensively used for network connectivity optimization by exploiting its diminishing returns property. Despite the empirical success, two key challenges largely remain open. First, on the theoretic side, the hardness, as well as the approximability of the general network connectivity optimization problem are still nascent except for a few special instances. Second, on the algorithmic side, current algorithms are often hard to balance between the optimization quality and the computational efficiency. In this paper, we systematically address these two challenges for the network connectivity optimization problem. First, we reveal some fundamental limits by proving that, for a wide range of network connectivity optimization problems, (1) they are NP-hard and (2) (1 − 1/e) is the optimal approximation ratio for any polynomial algorithms. Second, we propose an effective, scalable and general algorithm (CONTAIN) to carefully balance the optimization quality and the computational efficiency.where π is a subgraph of G, f is a non-negative function that maps any subgraph in G to a non-negative real number (i.e. f : π → R + ) [7]. Specifically, we have f (ϕ) = 0 for empty set ϕ; when f (π ) > 0, we call subgraph π as a valid subgraph. In other words, the network connectivity C(G) can be viewed as a weighted aggregation of the connectivities of all valid subgraphs in the network.By choosing an appropriate f () function (please refer to [7] for details), Eq. (1) includes several prevalent network connectivity measures, e.g., path capacity (which is in close relation to the epidemic threshold), triangle capacity (which is rooted in social balance theory) and natural connectivity (which is closely related to network robustness). In terms of computation, it is often much more efficient to either approximate or compute these connectivity measures by the associated eigen-function F (Λ (r ) ), where Λ (r ) represents the top-r eigenvalues of A. For example, the path capacity converges to the leading eigenvalue of the adjacency matrix of the network [4], the triangle capacity can be approximated by the sum of cubes of the eigenvalues [33], and the natural connectivity is calculated by the sum of exponentials of the eigenvalues [16]. Network Connectivity OptimizationWith the network connectivity measure in Eq. (1), we formally define network connectivity optimization problem as follows.Problem 1. Network Connectivity Optimization (NETCOP) Given: (1) a network G; (2) a connectivity mapping function f : π → R + which defines C(G); (3) a type of network operation (node deletion vs. edge deletion) and (4) a...