sxlxc/AGTslides
1
0
Fork 0
Code Issues Pull Requests Actions Packages Projects Releases Wiki Activity

...

Browse Source
This commit is contained in:
Yu Cong 2025-05-04 13:36:28 +08:00
parent 7f200d09c2
commit 883e95d3ee
2 changed files with 19 additions and 3 deletions
Show Stats Download Patch File Download Diff File Expand all files Collapse all files

5
beamerthemeSimple.sty
View File

@ -167,8 +167,9 @@
} }
% more theorem env % more theorem env
\newtheorem{observation}{Observation} \newtheorem{conjecture}[theorem]{Conjecture}
\newtheorem{question}{Question} \newtheorem{observation}[theorem]{Observation}
\newtheorem{question}[theorem]{Question}
% ---------------------------------------------------------------------- % ----------------------------------------------------------------------

17
main.tex
View File

@ -1,5 +1,6 @@
\documentclass{beamer} \documentclass{beamer}
\usepackage{nicefrac} \usepackage{nicefrac}
\DeclareMathOperator*{\opt}{OPT}
\title[Edge Conn Interdiction]{Faster FPTAS for Edge Connectivity Interdiction} \title[Edge Conn Interdiction]{Faster FPTAS for Edge Connectivity Interdiction}
\date{\today} \date{\today}
@ -78,7 +79,7 @@ The input is a graph $G=(V,E)$ with edge weights $w:E\to \Z_+$ and edge removal
Let $\tau$ be the optimum of Normalized Mincut. Consider a truncated weight $w_\tau(e)= \min \{w(e),c(e)\tau\}$. Let $\tau$ be the optimum of Normalized Mincut. Consider a truncated weight $w_\tau(e)= \min \{w(e),c(e)\tau\}$.
\begin{theorem} \begin{theorem}\label{thm:2approx}
The optimal cut $C^*$ for Connectivity Interdiction is a 2-approximation of global mincut with weights $w_\tau$. The optimal cut $C^*$ for Connectivity Interdiction is a 2-approximation of global mincut with weights $w_\tau$.
\end{theorem} \end{theorem}
\end{frame} \end{frame}
@ -168,7 +169,21 @@ s.t.& & \sum_{T\ni e} z_T &\leq w(e) & &\forall e\in E\\
Again we first assume the $\mu$ is fixed. Then for each pair of constraints $\sum z_T \leq w(e)$ and $\sum z_T \leq c(e)\mu$ only one of them works. The real capacity for this fractional tree packing is $\min\{w(e),c(e)\mu\}$, which is exactly the truncated weight $w_\tau$. Again we first assume the $\mu$ is fixed. Then for each pair of constraints $\sum z_T \leq w(e)$ and $\sum z_T \leq c(e)\mu$ only one of them works. The real capacity for this fractional tree packing is $\min\{w(e),c(e)\mu\}$, which is exactly the truncated weight $w_\tau$.
\end{frame} \end{frame}
\begin{frame}{Integrality Gap}
\begin{conjecture}
ILP\ref{IP} has an integrality gap of 4.
\end{conjecture}
Suppose $\mu^*$ is the optimal solution to LP\ref{lp:dualcutint}. Let $\lambda^{fr}$ be the fractional mincut with capacity $w_\mu$.
we have
\begin{align*}
4\opt(LP) &=4\lambda^{fr}-4b\mu^*\\
&\geq 2\lambda^{int} - 4b\mu^*\\
&\geq w_{\mu^*}(C^*)-b\mu^*,
\end{align*}
which implies Theorem \autoref{thm:2approx}.
\end{frame}
\begin{frame}{References} \begin{frame}{References}
\bibliographystyle{plainnat} \bibliographystyle{plainnat}
\bibliography{ref} \bibliography{ref}