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One can see that the optimal $\lambda$ is clearly 0 since $L(\lambda)$ is pwl concave and the slope is negative at $\lambda=0$. What we are really interested in is the first segment on $L$. At the left end, $L(0)$ is exactly the weight of minimum cut. (the complementary slackness condition is satisfied.) At the right end, as we have shown in the previous paragraph, $\lambda$ equals to the value of the strength (which is the optimum of the linear relaxation of the cut IP). However, for cut interdiction problems $L(0)$ is not the optimum. +\subsection{normalized mincut} +I read this trick in sparsest cut notes\footnote{\url{https://courses.grainger.illinois.edu/cs598csc/fa2024/Notes/lec-sparsest-cut.pdf}}. First we write a IP for normalized mincut. + +\begin{equation} +\begin{aligned}\label{ip:normalized} +\min& & \frac{\sum_e w(e)x_e}{b+1-\sum_e c(e)y_e}& & &\\ +s.t.& & \sum_{e\in T} x_e+y_e&\geq 1 & &\forall T\\ +& & \sum_{e} y_e c(e) &\leq b & &\\ +% & & x_e&\geq y_e & &\forall e\quad(F\subset C)\\ +& & y_e,x_e&\in\{0,1\} & &\forall e +\end{aligned} +\end{equation} + +A lp relaxation would be the following, +\begin{equation} +\begin{aligned}\label{lp:normalized} +\min& & \sum_e w(e)x_e& & &\\ +s.t.& & \sum_{e\in T} x_e+y_e&\geq 1 & &\forall T\\ +& & \sum_{e} y_e c(e) &= b & &\\ +% & & x_e&\geq y_e & &\forall e\quad(F\subset C)\\ +& & y_e,x_e&\geq 0 & &\forall e +\end{aligned} +\end{equation} +Note that we are forcing $b+1-\sum_e c(e)y_e=1$. + \subsection{integrality gap} I guess the 2-approximate min-cut enumeration algorithm implies an integrality gap of 2 for cut interdiction problem.