Based upon the idea that network functionality is impaired if two nodes in a network are sufficiently separated in terms of a given metric, we introduce two combinatorial \emph{pseudocut} problems generalizing the classical min-cut and multi-cut problems. We expect the pseudocut problems will find broad relevance to the study of network reliability. We comprehensively analyze the computational complexity of the pseudocut problems and provide three approximation algorithms for these problems. Motivated by applications in communication networks with strict Quality-of-Service (QoS) requirements, we demonstrate the utility of the pseudocut problems by proposing a targeted vulnerability assessment for the structure of communication networks using QoS metrics; we perform experimental evaluations of our proposed approximation algorithms in this context.