FPTAS section

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main.tex

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 \documentclass{beamer}
+\usepackage{nicefrac}
 
 \title[Edge Conn Interdiction]{Faster FPTAS for Edge Connectivity Interdiction}
 \date{\today}
@@ -42,37 +43,77 @@ Now suppose that we want to attack the network. To what extent can we decrease t
 \begin{problem}[edge connectivity interdiction]
 The input is a graph $G=(V,E)$ with edge weights $w:E\to \Z_+$ and edge removal cost $c:E\to \Z_+$ and a budget $b\in \Z_+$. The goal is to find a interdiction set $F\subset E$ with $c(F)\leq b$ that minimizes the mincut in $G-F$.
 \end{problem}
+
+How to solve this problem if\dots
+\begin{itemize}
+    \item the optimal $F$ is given?
+    \item the optimal $C$ is given?
+\end{itemize}
 \end{frame}
 
 
-\begin{frame}{Examples}
+\begin{frame}{Example}
 \begin{figure}
-Examples for containing knapsack and for unweighted easy case.
+\includegraphics[width=0.8\textwidth]{images/knapsack.png}
 \end{figure}
 \end{frame}
 
 \begin{frame}{Prevous Works}
 Zenklusen \citep{zenklusen_connectivity_2014} first studied this problem and showed the following results:
 \begin{itemize}
-\item A PTAS\footnote{polynomial time approximation scheme} for edge connectivity interdiction;
+\item A PTAS\footnote{polynomial time approximation scheme. The running time is polynomial in the input size if $\epsilon$ is fixed.} for edge connectivity interdiction;
 \item A $\tilde{O}(m^2 n^4)$ algorithm for the unit cost case\footnote{$\tilde{O}$ hides polylog factors}.
 \end{itemize}
 
-Later \citep{vygen_fptas_2024} discovered an FPTAS\footnote{fully PTAS} with time complexity $\tilde{O}(m^2 n^4/\epsilon)$.
+Later \citep{vygen_fptas_2024} discovered an FPTAS\footnote{Fully PTAS. The running time is polynomial in both the input size and $1/\epsilon$} with time complexity $\tilde{O}(m^2 n^4/\epsilon)$.
 \end{frame}
 
 
 \section{FPTAS}
 
-\begin{frame}{placeholder}
+\begin{frame}{Intermediate Problem}
+\begin{problem}[Normalized Mincut]
+The input is a graph $G=(V,E)$ with edge weights $w:E\to \Z_+$ and edge removal cost $c:E\to \Z_+$ and a budget $b\in \Z_+$. Find an edge set $F\subset E$ with $c(F)\leq b$ and a cut $C$ such that $\frac{w(C-F)}{b+1-c(F)}$ is minimized.
+\end{problem}
 
+Let $\tau$ be the optimum of Normalized Mincut. Consider a truncated weight $w_\tau(e)= \min \{w(e),c(e)\tau\}$.
+
+\begin{theorem}
+The optimal cut $C^*$ for Connectivity Interdiction is a 2-approximation of global mincut with weights $w_\tau$.
+\end{theorem}
+\end{frame}
+
+\begin{frame}{Algorithm}
+\begin{algo}
+\underline{\textsc{FPTAS for Connectivity Interdiction}}$(G,w,c,b)$\\
+1. estimate Normalized Mincut\\
+2. enumerate all 2-approximate mincut with weight $w_\tau$\\
+3. \quad for each cut $C$ solve a knapsack to compute $F$\\
+return $(C,F)$ with smallest objective value.
+\end{algo}
+
+1 takes $O(\log_{1+\epsilon}(poly(n)))$ time;\newline
+2 takes $O(n^4)$;\newline
+3 takes $O(m^2/\epsilon)$.
+\newline
+
+complexity: $\tilde{O}(m^2n^4/\epsilon)$.
 \end{frame}
 
 
 \section{LP Perspective}
 
-\begin{frame}{placeholder}
-
+\begin{frame}{LP Method}
+\citep{vygen_fptas_2024} gives a strong framework but the intuition behind is vague.
+\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 & &\text{($x+y$ is a cut)}\\
+&       &   \sum_{e} y_e c(e) &\leq b   & & &   &\text{(budget for $F$)}\\
+% &       &   x_e&\geq y_e   &   &\forall e\quad(F\subset C)\\
+&       &   y_e,x_e&\in\{0,1\}      &   &\forall e & &
+\end{aligned}
+\end{equation}
 \end{frame}
 
 \begin{frame}{References}

ref.bib

@@ -10,10 +10,8 @@
 	booktitle = {Integer {Programming} and {Combinatorial} {Optimization}},
 	publisher = {Springer Nature Switzerland},
 	author = {Huang, Chien-Chung and Obscura Acosta, Nidia and Yingchareonthawornchai, Sorrachai},
-	editor = {Vygen, Jens and Byrka, Jarosław},
 	year = {2024},
 	doi = {10.1007/978-3-031-59835-7_16},
-	note = {Series Title: Lecture Notes in Computer Science},
 	pages = {210--223},
 }
 
@@ -27,7 +25,5 @@
 	journal = {Operations Research Letters},
 	author = {Zenklusen, Rico},
 	year = {2014},
-	note = {Publisher: Elsevier B.V.},
-	keywords = {Approximation algorithms, Interdiction problems, Multi-objective optimization, Robust optimization},
 	pages = {450--454},
 }