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\title{Connectivity Interdiction \& \\ Perturbed Graphic Matroid Cogirth}
\date{\today}
\author{Cong Yu}
\institute{Algorithm \& Logic Group, UESTC}


\begin{document}
\maketitle
\section{Connectivity interdiction}
\begin{frame}{Connectivity interdiction}
\begin{problem}[connectivity interdiction {[Zenklusen ORL'14]}]
Let $G=(V,E)$ be a graph with edge weights $w:E\to\Z_+$ and edge removal costs $c:E\to\Z_+$ 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}

\pause\medskip
\begin{problem}[$B$-free min-cut]
Given the same input, define weight function for cuts
\[ w'(\delta(X))=\min_{F\subset \delta(X),c(F)\leq B} w(\delta(X)-F)\]

Find the cut with minimum weight $w'$.
\end{problem}
\end{frame}

\begin{frame}{Applications}
\begin{itemize}
\item drag delivery
\item nuclear smuggling
\item hospital infection control
\item ...
\end{itemize}
\end{frame}

\begin{frame}{Previous works}
\begin{table}[h]
\centering
\begin{tabular}{c c c c}
\toprule
unit cost $w(\cdot)=1$ & general case & random? & ref \\
\midrule
$O(m^2n^4\log n)$ & $O(m^{2+1/\e}n^4\log n)$ & $\times$  & [Zenklusen ORL'14] \\
$O(mn^4\log^2 n)$ & $O(m^{1+1/\e}n^4\log^2 n)$ & $\checkmark$ & [Zenklusen ORL'14] \\
$\tilde O(m+n^4 B)$ & $O(\frac{m^2n^4}{\e}\log(Bnw_{\max}))$ & $\times$ &   [Huang \etal{} IPCO'24]\\
$\tilde O(n^3m\log w_{\max})$ & - & $\checkmark$ &   [Drange \etal{} AAAI'26]\\
$\tilde O(m^2+n^3B)$ & $\tilde O(m^2+mn^3+\frac{n^3}{\e})$ & $\times$ &  this work\\
$\tilde O(m+n^3B)$ & $\tilde O(mn^3+\frac{n^3}{\e})$ & $\checkmark$ &  this work\\
\bottomrule
\end{tabular}
\caption{PTASes for connectivity interdiction}
\end{table}
\end{frame}

\begin{frame}{Method in [Huang \etal{} IPCO'24]}

\begin{problem}[\emph{normalized min-cut} {[Chalermsook \etal{} ICALP'22]}]
Given the same input as connectivity interdiction, find
\[\argmin_{\text{cut $C$}, F\subset C} \frac{w(C-F)}{B-c(F)+1} \quad s.t. \quad 0\leq c(F)\leq B.\]
\end{problem}

\pause\medskip
First considered in [Chalermsook \etal{} ICALP'22] as a subproblem in MWU framework when solving minimum $k$-edge connected spanning subgraph (ECSS) problem.
\end{frame}

\begin{frame}{Method in [Huang \etal{} IPCO'24]}
Let $\tau$ be a $(1+\e)$-approximation to the normalized min-cut and let $C^*$ be the optimal $B$-free cut.
\begin{lemma}
$C^*$ is a $(2+2\e)$-approximate min-cut in $(G,w_\tau)$, 

where $w_\tau(e)=\min(\tau c(e),w(e))$ is the truncated edge weight.
\end{lemma}
\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 \text{spanning tree $T$}\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}
\end{document}