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main.tex

@@ -35,8 +35,8 @@
 \pause
 
 \medskip
-\begin{problem}[Zenklusen ORL'14]
-Let $G=(V,E)$ be a graph with edge capacity $c:E\to\mathbb{Z}_+$ and edge removal costs $w:E\to\mathbb{Z}_+$ and let $B\in \mathbb Z_+$ be the budget.
+\begin{problem}[{[Zenklusen ORL'14]}]
+Let $G=(V,E)$ be a graph with edge weights $w:E\to\mathbb{Z}_+$ and edge removal costs $c:E\to\mathbb{Z}_+$ and let $B\in \mathbb Z_+$ be the budget.
 
 Find $F\subset E$ with $w(F)\leq B$ s.t. the min-cut in $G-F$ is minimized.
 \end{problem}
@@ -57,8 +57,8 @@ unit cost $w(\cdot)=1$ & general case & random? & ref \\
 \midrule
 $O(m^2n^4\log n)$ & $O(m^{2+1/\varepsilon}n^4\log n)$ & $\times$  & [Zenklusen ORL'14] \\
 $O(mn^4\log^2 n)$ & $O(m^{1+1/\varepsilon}n^4\log^2 n)$ & $\checkmark$ & [Zenklusen ORL'14] \\
-$\tilde O(m+n^4 B)$ & $O(\frac{m^2n^4}{\varepsilon}\log(Bnc_{\max}))$ & $\times$ &   [C.-C. Huang et al. IPCO'24]\\
-$\tilde O(n^3m\log c_{\max})$ & - & $\checkmark$ &   [Drange et al. AAAI'26]\\
+$\tilde O(m+n^4 B)$ & $O(\frac{m^2n^4}{\varepsilon}\log(Bnw_{\max}))$ & $\times$ &   [C.-C. Huang et al. IPCO'24]\\
+$\tilde O(n^3m\log w_{\max})$ & - & $\checkmark$ &   [Drange et al. AAAI'26]\\
 $\tilde O(m^2+n^3B)$ & $\tilde O(m^2+mn^3+\frac{n^3}{\varepsilon})$ & $\times$ &  this work\\
 $\tilde O(m+n^3B)$ & $\tilde O(mn^3+\frac{n^3}{\varepsilon})$ & $\checkmark$ &  this work\\
 \bottomrule
@@ -66,6 +66,31 @@ $\tilde O(m+n^3B)$ & $\tilde O(mn^3+\frac{n^3}{\varepsilon})$ & $\checkmark$ &
 \caption{PTASes for connectivity interdiction}
 \end{table}
 \end{frame}
+
+\begin{frame}{Method in [C.-C. Huang et al. IPCO'24]}
+They use an intemediate problem called \emph{normalized min-cut}.
+\begin{problem}[{[Chalermsook ICALP'22]}]
+Given the same input as connectivity interdiction, find a cut $C$ and a subset of its edges $F\subset C$ satisfying $0\leq c(F)\leq B$, so that $\frac{w(C-F)}{B-c(F)+1}$ is minimized.
+\end{problem}
+
+\pause\medskip
+Normalized min-cut is first considered in [Chalermsook ICALP'22] as a subproblem in MWU framework when solving minimum $k$-edge connected spanning subgraph (ECSS) problem.
+
+\pause\medskip
+...
+\end{frame}
+
+\begin{frame}{LP method}
+\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{($x+y$ is a cut)}\\
+&       &   \sum_{e} y_e c(e) &\leq B       &   &\text{(budget for $F$)}\\
+&       &   y_e,x_e&\in\{0,1\}              &   &\forall e
+\end{aligned}
+\end{equation*}
+\end{frame}
+
 \section{Computing cogirth in perturbed graphic matroids}
 \begin{frame}{Perturbed graphic matroids}
 \end{frame}