Client-server networks are pervasive, fundamental, and include such key networks as the Internet, power grids, and road networks. In a client-server network, clients obtain a service by connecting to one of a redundant set of servers. These networks are vulnerable to node and link failures, causing some clients to become disconnected from the servers. We develop algorithms that quantify and bound the inherent vulnerability of a client-server network using semidefinite programming (SDP) and branch-and-cut techniques. Further, we develop a divide-and-conquer algorithm that solves the problem for large graphs. We use these techniques to show that: for the Philippine Power Grid removing just over 6% of the transmission lines will disconnect at least 20% but not more than 50% of the substations from all generators; on a large wireless mesh network disrupting 5% of wireless links between relays removes Internet access for half the relays; even after any 16% of Tier 2 ASes are removed, more than 50% of the remaining Tier 2 ASes will be connected to the Tier 1 backbone; when 300 roadblocks are erected in Michigan, it's possible to disconnect 28-43% of the population from all airports.
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