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@@ -46,13 +46,13 @@
\begin{problem}[connectivity interdiction {[Zenklusen, ORL'14]}] \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. 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. Find $F\subset E$ with $c(F)\leq B$ s.t. the edge connectivity in $G-F$ is minimized.
\end{problem} \end{problem}
\p \p
\begin{problem}[$B$-free min-cut] \begin{problem}[$B$-free min-cut]
Given the same input, define weight function for cuts Given the same input, define weight function for cuts
\[ w'(\delta(X))=\min_{F\subset \delta(X),c(F)\leq B} w(\delta(X)-F)\] \[ w'(\delta(X))=\min_{\substack{F\subset \delta(X)\\ c(F)\leq B}} w(\delta(X)-F)\]
Find the cut with minimum weight $w'$. Find the cut with minimum weight $w'$.
\end{problem} \end{problem}
@@ -68,22 +68,25 @@ Find the cut with minimum weight $w'$.
\end{frame} \end{frame}
\begin{frame}{Previous works} \begin{frame}{Previous works}
\newcommand{\blueT}{\textcolor{blue}{T}}
\begin{table}[h] \begin{table}[h]
\centering \centering
\begin{tabular}{c c c c} \begin{tabular}{c c c c}
\toprule \toprule
unit cost $w(\cdot)=1$ & general case & random? & ref \\ Unit cost $\textcolor{gray}{c(\cdot)=1}$ & General case & Random? & Reference \\
\midrule \midrule
$O(m^2n^4\log n)$ & $O(m^{2+1/\e}n^4\log n)$ & $\times$ & [Zenklusen, ORL'14] \\ $O(m^2n^4\log n)$ & $O(m^{2+1/\e}n^4\log n)$ & $\times$ & [Zenklusen, ORL'14] \\
$\tilde O(m+n^4 B)$ & $O(n^4\log(Bnw_{\max})\blueT)$ & $\times$ & [Huang \etal{}, IPCO'24]\\
$\tilde O(m^2+n^3B)$ & $\tilde O(m^2+n^3\blueT)$ & $\times$ & this work\\
\midrule
$O(mn^4\log^2 n)$ & $O(m^{1+1/\e}n^4\log^2 n)$ & $\checkmark$ & [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]\\ $\tilde O(mn^3\log w_{\max})$ & & $\checkmark$ & [Drange \etal{}, AAAI'26]\\
$\tilde O(mn^3\log w_{\max})$ & - & $\checkmark$ & [Drange \etal{}, AAAI'26]\\ $\tilde O(m+n^3B)$ & $\tilde O(m+n^3\blueT)$ & $\checkmark$ & this work\\
$\tilde O(m^2+n^3B)$ & $\tilde O(m^2+mn^3+n^3/\e)$ & $\times$ & this work\\
$\tilde O(m+n^3B)$ & $\tilde O(mn^3+n^3/\e)$ & $\checkmark$ & this work\\
\bottomrule \bottomrule
\end{tabular} \end{tabular}
\caption{PTASes for connectivity interdiction} \caption{PTASes for connectivity interdiction}
\end{table} \end{table}
$\blueT$ is the complexity of FPTAS for 0-1 knapsack.
\end{frame} \end{frame}
\begin{frame}{Method in [Huang \etal{}, IPCO'24]} \begin{frame}{Method in [Huang \etal{}, IPCO'24]}