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gap2

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Yu Cong 2025-05-06 11:29:17 +08:00
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beamerthemeSimple.sty
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main.tex
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\documentclass{beamer}
\usepackage{nicefrac}
\DeclareMathOperator*{\opt}{OPT}
\title[Edge Conn Interdiction]{Faster FPTAS for Edge Connectivity Interdiction}
@ -158,7 +157,7 @@ $\mu_i=\min \frac{w(C\setminus F)-w(C_{i-1}\setminus F_{i-1})}{c(F_{i-1})-c(F)}$
\begin{frame}{Weight Truncation from LP}
Consider the dual of linear relaxation of ILP\ref{IP}.
Consider the dual of the linear relaxation of IP\ref{IP}.
\begin{equation}\label{lp:dualcutint}
\begin{aligned}
@ -173,19 +172,14 @@ Again we first assume the $\mu$ is fixed. Then for each pair of constraints $\su
\end{frame}
\begin{frame}{Integrality Gap}
% \begin{conjecture}
% ILP\ref{IP} has an integrality gap of 4.
% \end{conjecture}
\begin{conjecture}\label{conj:gap2}
IP\ref{IP} has an integrality gap of 2.
\end{conjecture}
% Suppose $\mu^*$ is the optimal solution to LP\ref{lp:dualcutint}. Let $\lambda^{fr}$ be the fractional mincut with capacity $w_\mu$.
Suppose that $\mu^*$ is the optimal solution to LP\ref{lp:dualcutint}. Let $\lambda^{fr}$ and $\lambda^{int}$ be the fractional and integral mincut with capacity $w_{\mu^*}$.
\newline
% 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}.
Conjecture \ref{conj:gap2} implies $w_{\mu^*}(C^*)+b\mu^* \geq (\text{value of mincut with $w_{\mu^*}$})$, \newline which is stronger than Theorem \ref{thm:2approx}.
\end{frame}
\begin{frame}{References}
\bibliographystyle{plainnat}