83 lines
2.3 KiB
TeX
83 lines
2.3 KiB
TeX
\documentclass{beamer}
|
|
|
|
\title[Edge Conn Interdiction]{Faster FPTAS for Edge Connectivity Interdiction}
|
|
\date{\today}
|
|
\author{丛宇}
|
|
% \AtBeginSection[]{
|
|
% \frame{\frametitle{Outline}\tableofcontents[currentsection,
|
|
% subsectionstyle=show/show/shaded]}
|
|
% }
|
|
|
|
\usetheme{Simple}
|
|
\usepackage{algo}
|
|
|
|
\begin{document}
|
|
\begin{frame}[plain]
|
|
% Print the title page as the first slide
|
|
\titlepage
|
|
\end{frame}
|
|
|
|
\begin{frame}[plain]{Plan}
|
|
\tableofcontents
|
|
\end{frame}
|
|
|
|
\section{Connectivity Interdiction}
|
|
|
|
\begin{frame}{Edge Connectivity \& Cut}
|
|
\begin{definition}
|
|
Let $G=(V,E)$ be a undirected connected graph. The minimum cut of $G$ is the minimum set of edges whose removal breaks the connectivity of $G$.
|
|
\end{definition}
|
|
|
|
The edge connectivity of $G$ = $|\text{mincut of $G$}|-1$.
|
|
\newline
|
|
|
|
Edge connectivity is an important measure of network reliability. The greater the edge connectivity, the more difficult it is to break the network's connectivity.
|
|
\newline
|
|
|
|
Now suppose that we want to attack the network. To what extent can we decrease the size of the min-cut by removing a limited set of edges?
|
|
\end{frame}
|
|
|
|
|
|
\begin{frame}{Interdiction}
|
|
\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}
|
|
\end{frame}
|
|
|
|
|
|
\begin{frame}{Examples}
|
|
\begin{figure}
|
|
Examples for containing knapsack and for unweighted easy case.
|
|
\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 $\tilde{O}(m^2 n^4)$ 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 $\tilde{O}(m^2 n^4/\epsilon)$.
|
|
\end{frame}
|
|
|
|
|
|
\section{FPTAS}
|
|
|
|
\begin{frame}{placeholder}
|
|
|
|
\end{frame}
|
|
|
|
|
|
\section{LP Perspective}
|
|
|
|
\begin{frame}{placeholder}
|
|
|
|
\end{frame}
|
|
|
|
\begin{frame}{References}
|
|
\bibliographystyle{plainnat}
|
|
\bibliography{ref}
|
|
\end{frame}
|
|
|
|
\end{document} |