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minimum cost of cuts in $G$. We have $-\lambda(b-B)\leq w(C\setminus F)-\lambda(b-c(F))$ for any cut $C$ and $F\subsetneq C$. Thus the upperbound is $\e=\min \frac{w(C\setminus F)}{B-c(F)}$. -\note{It remains to show that the optimal solution at $\e$ guarantees $c(F)\leq b$? or maybe we don't need this for normalized mincut.} +\note{It remains to show that the optimal solution at $\e$ guarantees $c(F)\leq b$? or maybe we don't need this for normalized mincut. I think normalized min-cut should not require $c(F)\leq b$. Further checks are needed. What we can guarantee is that $c(F)\leq B$.} \subsection{integrality gap} I guess the 2-approximate min-cut enumeration algorithm implies an integrality gap of 2 for cut interdiction problem. @@ -245,6 +245,10 @@ We want to prove something like tree packing theorem for \autoref{lp:dualcutint} 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} +\begin{conjecture} + \autoref{lp:cutinterdict} has an integrality gap of 2. +\end{conjecture} + \section{Random Stuff} \subsection{remove box constraints}