diff --git a/beamerthemeSimple.sty b/beamerthemeSimple.sty index 293ad81..a3f3fca 100644 --- a/beamerthemeSimple.sty +++ b/beamerthemeSimple.sty @@ -141,7 +141,7 @@ \setbeamercolor{alerted text}{fg=beamer@simple@color} \setbeamerfont{block title alerted}{series=\mdseries} \setbeamerfont{alerted text}{series=\bfseries\boldmath} -\hypersetup{colorlinks=true,linkcolor=,citecolor=Green,urlcolor=oliver} +\hypersetup{colorlinks=true,linkcolor=oliver,citecolor=Green,urlcolor=oliver} % \usefonttheme[onlymath]{serif} \setbeamerfont{frametitle}{series=\bfseries\boldmath} \setbeamerfont{block title}{series=\bfseries\boldmath} diff --git a/main.tex b/main.tex index a08c60d..99af996 100644 --- a/main.tex +++ b/main.tex @@ -1,5 +1,4 @@ \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}