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--- a/main.tex
+++ b/main.tex
@@ -1,5 +1,25 @@
 \documentclass[12pt]{article}
 \usepackage{chao}
+\usepackage[breakable, theorems, skins]{tcolorbox}
+\tcbset{enhanced}
+\DeclareRobustCommand{\note}[2][blue]{%
+\begin{tcolorbox}[
+        breakable,
+        left=0pt,
+        right=0pt,
+        top=0pt,
+        bottom=0pt,
+        colback=white,
+        colframe=#1,
+        width=\dimexpr\textwidth\relax, 
+        enlarge left by=0mm,
+        boxsep=5pt,
+        arc=0pt,outer arc=0pt,
+        ]
+        #2
+\end{tcolorbox}
+}
+
 \title{connectivity interdiction}
 \author{}
 \date{}
@@ -200,7 +220,12 @@ We are interested in the upperbound $\e$ of $\lambda$ such that the optimal $F$
 % (there is a $\pm1$ difference between principal partition and graph strength... but we dont care those $c\lambda$ terms since the difficult part is minimize $L(\lambda)$ for fixed $\lambda$)
 
 \subsection{principal sequence of partitions for cut interdiction}
-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 $G$.
+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 $c$. Note that 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 $G$.
+Let $B$ be the 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.}
 
 \subsection{integrality gap}
 I guess the 2-approximate min-cut enumeration algorithm implies an integrality gap of 2 for cut interdiction problem.
@@ -215,7 +240,7 @@ s.t.&       &       \sum_{T\ni e} z_T &\leq w(e)    &   &\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}.
+We want to prove something like tree packing theorem 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}