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A further reformulation (the new $x$ is $x-y$) gives us the following, -\begin{equation*} +\begin{equation}\label{lp:cutinterdict} \begin{aligned} \min& & \sum_{e} x_e w(e) & & \\ s.t.& & \sum_{e\in T} x_e+y_e&\geq 1 & &\forall T\quad \text{($x+y$ is a cut)}\\ @@ -162,7 +162,7 @@ s.t.& & \sum_{e\in T} x_e+y_e&\geq 1 & &\forall T\quad \text{($x+y$ % & & x_e&\geq y_e & &\forall e\quad(F\subset C)\\ & & y_e,x_e&\in\{0,1\} & &\forall e \end{aligned} -\end{equation*} +\end{equation} Note that now this is almost a positive covering LP. Let $L(\lambda)= \min \{ w(C\setminus F)-\lambda(b-c(F)) | \forall \text{cut $C$}\;\forall F\subset C % \land c(F)\leq b @@ -203,7 +203,22 @@ We are interested in the upperbound $\e$ of $\lambda$ such that the optimal $F$ Now we focus on $L(\lambda)=\min \{w(C\setminus F)-\lambda(b-c(F)) | \forall \text{cut } C\;\forall F\subset C\}$. We can still assume that $G$ is connected and see that $L(\lambda)$ is pwl concave (1 and 2 still hold). Let $\lambda^*$ be a breakpoint on $L$. Suppose that there are two optimal solutions $(C_1,F_1)$ and $(C_2,F_2)$ at $\lambda^*$. For fixed $C$ ($C_1=C_2$), the same argument for principal partition still works. However, the difficult part is that $C$ might not be the same. So it's unlikely that 3 and 4 hold. For cut interdiction problem, 5 shows connections between normalized mincut and the original interdiction problem. Recall that we observe the denominator in normalized min-cut can be relaxed (that is, we can use $\frac{w(C\setminus F)}{B-c(F)}$ for any $B>b$, instead of restricting to $B=b+1$) and the analysis still works. Now following the previous argument for 5, we assume $\lambda\in [0,\e]$ for small enough positive $\e$. For any $C$, we have $F=C$ since $w(C\setminus F)$ is dominating. For the remaining term $-\lambda(b-c(F))$ we are selecting a cut $F$ with smallest cose with respect to $c$. We can assume that any cut in $G$ has larger cost than $b$ since otherwise the optimum is simply 0. Now we can see that $B$ in the denominator $B-c(F)$ should be the cost of mincut in $G$. \subsection{integrality gap} -I guess the 2-approximate min-cut enumeration algorithm implies a integrality gap of 2 for cut interdiction problem. +I guess the 2-approximate min-cut enumeration algorithm implies an integrality gap of 2 for cut interdiction problem. + +First consider the dual of linear relaxation of \autoref{lp:cutinterdict}. + +\begin{equation}\label{lp:dualcutint} +\begin{aligned} +\max& & \sum_T z_T &- b\lambda & &\\ +s.t.& & \sum_{T\ni e} z_T &\leq w(e) & &\forall e\in E\\ + & & \sum_{T\ni e} z_T &\leq c(e)\lambda & &\forall e \in E\\ + & & z_T,\lambda &\geq 0 & & +\end{aligned} +\end{equation} +We want to prove something like tree packing for \autoref{lp:dualcutint}. +\begin{conjecture} + The optimum of \autoref{lp:dualcutint} is $\min \set{\frac{w(C\setminus F)}{B-c(F)}| \forall \text{cut $C$}, c(F)\leq b}$, where $B$ is the cost of mincut in $G$ and $b$ is the budget. +\end{conjecture} \section{Random Stuff}