z

main.tex

@@ -5,6 +5,7 @@
 \documentclass[aspectratio=169,handout]{beamer}
 \usefonttheme{professionalfonts}
 
+\usepackage{algo}
 \usetheme{moloch}   % new metropolis
 % fonts
 \usepackage{fontspec}
@@ -20,6 +21,20 @@
 \usepackage{lete-sans-math}
 
 \usepackage{booktabs}
+\def\etal{\emph{et~al.}}
+\def\Real{\mathbb{R}}
+\def\Proj{\mathbb{P}}
+\def\Hyper{\mathbb{H}}
+\def\Integer{\mathbb{Z}}
+\def\Natural{\mathbb{N}}
+\def\Complex{\mathbb{C}}
+\def\Rational{\mathbb{Q}}
+\let\N\Natural
+\let\Q\Rational
+\let\R\Real
+\let\Z\Integer
+\def\Rd{\Real^d}
+\newcommand{\e}{\varepsilon}
 
 \title{Connectivity Interdiction \& \\ Perturbed Graphic Matroid Cogirth}
 \date{\today}
@@ -31,23 +46,30 @@
 \maketitle
 \section{Connectivity interdiction}
 \begin{frame}{Connectivity interdiction}
-\textit{What is the maximum change of edge connectivity in a network when some limited set of edges is being removed?}
-\pause
-
-\medskip
-\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.
+\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
-connectivity interdiction $\approx$ min-cut + knapsack
-
-Is there an FPTAS?
+\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
@@ -55,36 +77,43 @@ Is there an FPTAS?
 \toprule
 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] \\
-$\tilde O(m+n^4 B)$ & $O(\frac{m^2n^4}{\varepsilon}\log(Bnw_{\max}))$ & $\times$ &   [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\\
+$O(m^2n^4\log n)$ & $O(m^{2+1/\e}n^4\log n)$ & $\times$  & [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 [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.
+\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
-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.
+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}
 
-\pause\medskip
-...
+\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 T\quad \text{($x+y$ is a cut)}\\
+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}