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The assessment of network vulnerability is of great importance in the presence of unexpected disruptive events or adversarial attacks targeting on critical network links and nodes. In this paper, we study Critical Link Disruptor (CLD) and Critical Node Disruptor (CND) optimization problems to identify critical links and nodes in a network whose removals maximally destroy the network's functions. We provide a comprehensive complexity analysis of CLD and CND on general graphs and show that they still remain NP-complete even on unit disk graphs and power-law graphs. Furthermore, the CND problem is shown NP-hard to be approximated within Ω([(n-k)/(nε)] ) on general graphs with n vertices and k critical nodes. Despite the intractability of these problems, we propose HILPR, a novel LP-based rounding algorithm, for efficiently solving CLD and CND problems in a timely manner. The effectiveness of our solutions is validated on various synthetic and real-world networks.