generated from sxlxc/pdflatex-note
126 lines
3.9 KiB
TeX
126 lines
3.9 KiB
TeX
\RequirePackage{scrlfile}
|
|
\makeatletter
|
|
\AfterPackage{beamerbasemodes}{\beamer@amssymbfalse}
|
|
\makeatother
|
|
\documentclass[aspectratio=169,handout]{beamer}
|
|
\usefonttheme{professionalfonts}
|
|
|
|
\usepackage{algo}
|
|
\usetheme{moloch} % new metropolis
|
|
% fonts
|
|
\usepackage{fontspec}
|
|
\setsansfont[
|
|
ItalicFont={Fira Sans Italic},
|
|
BoldFont={Fira Sans Medium},
|
|
BoldItalicFont={Fira Sans Medium Italic}
|
|
]{Fira Sans}
|
|
\setmonofont[BoldFont={Fira Mono Medium}]{Fira Mono}
|
|
\AtBeginEnvironment{tabular}{%
|
|
\addfontfeature{Numbers={Monospaced}}
|
|
}
|
|
\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}
|
|
\author{Cong Yu}
|
|
\institute{Algorithm \& Logic Group, UESTC}
|
|
|
|
|
|
\begin{document}
|
|
\maketitle
|
|
\section{Connectivity interdiction}
|
|
\begin{frame}{Connectivity interdiction}
|
|
\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
|
|
\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
|
|
\begin{tabular}{c c c c}
|
|
\toprule
|
|
unit cost $w(\cdot)=1$ & general case & random? & ref \\
|
|
\midrule
|
|
$O(m^2n^4\log n)$ & $O(m^{2+1/\e}n^4\log n)$ & $\times$ & [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 [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
|
|
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}
|
|
|
|
\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 \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}
|
|
\end{equation*}
|
|
\end{frame}
|
|
|
|
\section{Computing cogirth in perturbed graphic matroids}
|
|
\begin{frame}{Perturbed graphic matroids}
|
|
\end{frame}
|
|
\end{document} |