sxlxc/edge_conn_interdiction
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Yu Cong 2025-04-19 14:26:12 +08:00
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commit 46771bae89
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@ -131,7 +131,7 @@ How to derive normalized min cut for connectivity interdiction?
\begin{aligned} \begin{aligned}
\max& & z& & & \\ \max& & z& & & \\
s.t.& & \sum_{e} y_e c(e) &\leq B & &\text{(budget for $F$)}\\ 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 & &\\ & & \sum_{e} \min(0,x_e-y_e) w(e)&\geq z & &\\
& & y_e,x_e&\in\{0,1\} & &\forall e & & y_e,x_e&\in\{0,1\} & &\forall e
\end{aligned} \end{aligned}
@ -143,7 +143,7 @@ we can assume that $y_e\leq x_e$.
\begin{aligned} \begin{aligned}
\min& & \sum_{e} (x_e&-y_e) w(e) & & \\ \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} (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 $F$)}\\ & & \sum_{e} y_e c(e) &\leq B & &\text{(budget for $F$)}\\
& & x_e&\geq y_e & &\forall e\quad(F\subset C)\\ & & x_e&\geq y_e & &\forall e\quad(F\subset C)\\
& & y_e,x_e&\in\{0,1\} & &\forall e & & 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{equation*}
\begin{aligned} \begin{aligned}
\min& & \sum_{e} x_e w(e) & & \\ \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 $F$)}\\ & & \sum_{e} y_e c(e) &\leq B & &\text{(budget for $F$)}\\
% & & x_e&\geq y_e & &\forall e\quad(F\subset C)\\ % & & x_e&\geq y_e & &\forall e\quad(F\subset C)\\
& & y_e,x_e&\in\{0,1\} & &\forall e & & y_e,x_e&\in\{0,1\} & &\forall e