diff --git a/images/L.png b/images/L.png new file mode 100644 index 0000000..f130c25 Binary files /dev/null and b/images/L.png differ diff --git a/main.tex b/main.tex index aaa0e3f..0fb8a1e 100644 --- a/main.tex +++ b/main.tex @@ -137,6 +137,13 @@ Thus the optimal solution must be $C=F=\text{mincut with capacity $c$}$. \end{frame} +\begin{frame}{Plot $L(\mu)$} +\begin{figure} +\includegraphics[width=0.8\linewidth]{images/L.png} +\end{figure} +\end{frame} + + \begin{frame}{Breakpoints on $L(\mu)$} We have see that the first line segment is $L(\mu)=(\lambda_c-b)\mu$ where $\lambda_c$ is the value of mincut with capacity $c$. @@ -176,7 +183,8 @@ Again we first assume the $\mu$ is fixed. Then for each pair of constraints $\su IP\ref{IP} has an integrality gap of 2. \end{conjecture} -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 $w_{\mu^*}$. +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 Conjecture \ref{conj:gap2} implies $w_{\mu^*}(C^*)+b\mu^* \leq 2(\text{value of mincut with $w_{\mu^*}$})$, \newline which is stronger than Theorem \ref{thm:2approx}.