In this chapter, we investigate the restricted inverse optimal value problem on the maximum capacity path under weighted \(l_1\) and \(l_{\infty }\) norms and address the interdiction problem with bounded capacity adjustments under the \(l_1\) norm. Without altering the capacity values of given paths, we modify the algorithm (AIMCPM) proposed in Deaconu and Tayyebi (IEEE Access 8:225957–225966;2020). For these problems, we first present the mathematical models, then design solution algorithms, prove the optimality of the algorithms, and analyze their time complexity. The interdiction problem is addressed with a binary search method, resulting in an \(O(m^2\log m)\) time complexity algorithm, which greatly improved the algorithm’s complexity proposed in Deaconu and Tayyebi (IEEE Access 8:225957–225966;2020). For the restricted inverse optimal value problem, it is tackled with a minimum cost cut approach in \(O(mn)\) time under \(l_1\) norm, while by finding maximum capacity paths in \(O(m)\) time under \(l_{\infty }\) norm.