normalized min-cut -- Lagrangian dual

main.tex

@@ -152,7 +152,14 @@ s.t.&       &   \sum_{e\in T} x_e+y_e&\geq 1   &   &\forall T\quad \text{($x$ fo
 \end{aligned}
 \end{equation*}
 
-Note that now this is almost a positive covering LP.
+Note that now this is almost a positive covering LP. Let $L(\lambda)= \min \{ w(C\setminus F)-\lambda(b-c(F)) | \forall \text{cut $C$}\;\forall F\subset C \land c(F)\leq b\}$ Consider the Lagrangian dual,
+\begin{equation*}
+\max_{\lambda\geq 0} L(\lambda)=  \max_{\lambda\geq 0} \min \left\{ w(C\setminus F)-\lambda(b-c(F)), \forall \text{cut $C$}\;\forall F\subset C \land c(F)\leq b\right\}
+\end{equation*}
+
+
+At this point, it becomes clear how the normalized min-cut is implicated in \cite{vygen_fptas_2024}. The optimum of normalized min-cut is exactly the value of $\lambda$ when $L(\lambda)$ is 0.
+
 
 \section{Random Stuff}