z

main.tex

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