lp for normalized mincut

main.tex

@@ -233,6 +233,31 @@ We have $-\lambda(b-B)\leq w(C\setminus F)-\lambda(b-c(F))$ for any cut $C$ and
 \subsection{differences}
 Consider $L(\lambda)$ for cut problem. One can see that the optimal $\lambda$ is clearly 0 since $L(\lambda)$ is pwl concave and the slope is negative at $\lambda=0$. What we are really interested in is the first segment on $L$. At the left end, $L(0)$ is exactly the weight of minimum cut. (the complementary slackness condition is satisfied.) At the right end, as we have shown in the previous paragraph, $\lambda$ equals to the value of the strength (which is the optimum of the linear relaxation of the cut IP). However, for cut interdiction problems $L(0)$ is not the optimum.
 
+\subsection{normalized mincut}
+I read this trick in sparsest cut notes\footnote{\url{https://courses.grainger.illinois.edu/cs598csc/fa2024/Notes/lec-sparsest-cut.pdf}}. First we write a IP for normalized mincut.
+
+\begin{equation}
+\begin{aligned}\label{ip:normalized}
+\min&   &   \frac{\sum_e w(e)x_e}{b+1-\sum_e c(e)y_e}&        &   &\\
+s.t.&       &   \sum_{e\in T} x_e+y_e&\geq 1   &   &\forall T\\
+&       &   \sum_{e} y_e c(e) &\leq b   &   &\\
+% &       &   x_e&\geq y_e   &   &\forall e\quad(F\subset C)\\
+&       &   y_e,x_e&\in\{0,1\}      &   &\forall e
+\end{aligned}
+\end{equation}
+
+A lp relaxation would be the following,
+\begin{equation}
+\begin{aligned}\label{lp:normalized}
+\min&   &   \sum_e w(e)x_e&        &   &\\
+s.t.&       &   \sum_{e\in T} x_e+y_e&\geq 1   &   &\forall T\\
+&       &   \sum_{e} y_e c(e) &= b   &   &\\
+% &       &   x_e&\geq y_e   &   &\forall e\quad(F\subset C)\\
+&       &   y_e,x_e&\geq 0     &   &\forall e
+\end{aligned}
+\end{equation}
+Note that we are forcing $b+1-\sum_e c(e)y_e=1$.
+
 \subsection{integrality gap}
 I guess the 2-approximate min-cut enumeration algorithm implies an integrality gap of 2 for cut interdiction problem.