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zRZ$ei@x9KzTA6w)ds%5uZ)NwU<*SG9(jHotT2}XJnNouY1fAKaNw8J6wlr za2Yzy)z@l87j(k~*6K@Mg&T0iT=zdc*9SLY5Qbow75cTe;5LlF9k|O~{r!88hXRb6 z8<^ClV`fdymF`gN3(|k540?vym^2k)?&m*T<1>bCK`ABR_cf z=WOKH%gFMKc46RxCPx#&90cYdFb6?%Py-qBe+qHgO3tkV`cqUJm)7h9APF@$3MC~) GPeuxbix$TK diff --git a/main.tex b/main.tex index eaf0de5..3a887cf 100644 --- a/main.tex +++ b/main.tex @@ -131,7 +131,7 @@ How to derive normalized min cut for connectivity interdiction? \begin{aligned} \max& & z& & & \\ s.t.& & \sum_{e} y_e c(e) &\leq B & &\text{(budget for $F$)}\\ -& & \sum_{e\in T} x_e&\geq 1 & &\forall T\quad \text{($x$ forms a cut)}\\ +& & \sum_{e\in T} x_e&\geq 1 & &\forall T\quad \text{($x$ is a cut)}\\ & & \sum_{e} \min(0,x_e-y_e) w(e)&\geq z & &\\ & & y_e,x_e&\in\{0,1\} & &\forall e \end{aligned} @@ -143,7 +143,7 @@ we can assume that $y_e\leq x_e$. \begin{aligned} \min& & \sum_{e} (x_e&-y_e) w(e) & & \\ % s.t.& & \sum_{e} (x_e-y_e) w(e)&\geq z & &\\ -s.t.& & \sum_{e\in T} x_e&\geq 1 & &\forall T\quad \text{($x$ forms a cut)}\\ +s.t.& & \sum_{e\in T} x_e&\geq 1 & &\forall T\quad \text{($x$ is a cut)}\\ & & \sum_{e} y_e c(e) &\leq B & &\text{(budget for $F$)}\\ & & x_e&\geq y_e & &\forall e\quad(F\subset C)\\ & & y_e,x_e&\in\{0,1\} & &\forall e @@ -157,7 +157,7 @@ A further reformulation (the new $x$ is $x-y$) gives us the following, \begin{equation*} \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{($x$ forms a cut)}\\ +s.t.& & \sum_{e\in T} x_e+y_e&\geq 1 & &\forall T\quad \text{($x+y$ is a cut)}\\ & & \sum_{e} y_e c(e) &\leq B & &\text{(budget for $F$)}\\ % & & x_e&\geq y_e & &\forall e\quad(F\subset C)\\ & & y_e,x_e&\in\{0,1\} & &\forall e