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@@ -140,6 +140,20 @@ s.t.&       &   \sum_{e\in T} x_e&\geq 1   &   &\forall T\quad \text{($x$ forms
 
 Now this LP looks similar to the normalized min-cut problem.
 
+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)}\\
+&       &   \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
+\end{aligned}
+\end{equation*}
+
+Note that now this is almost a positive covering LP.
+
 \section{Random Stuff}
 
 \subsection{remove box constraints}