counterexample for gap conj

main.tex

@@ -188,8 +188,31 @@ Suppose that $\mu^*$ is the optimal solution to LP\ref{lp:dualcutint}.
 \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}.
+\newline
+
+However, computational experiments show that the integrality gap of IP\ref{IP} is not a constant.
 \end{frame}
 
+
+\begin{frame}{Counterexample}
+Consider a cycle $C_n$ of $n$ vertices with two special edges $e_1,e_2$. Let $L$ be a large number.
+\[
+w(e)=\begin{cases}
+    1 & e=e_1\\
+    L & e=e_2\\
+    2 & \text{else}
+\end{cases},\quad
+c(e)=\begin{cases}
+    L & e=e_1\\
+    1 & \text{else}
+\end{cases}, \quad b=2-\epsilon
+\]
+
+For IP, it is clear that $F=\{e_2\}, C\setminus F=\{e_1\}$ and the optimum is 1\newline
+For LP, we assign $x=0$ and $y_e=\frac{1}{n-2}$ for every edge except $e_1$. The optimum is 0.
+\end{frame}
+
+
 \begin{frame}{Gaps}
 \vspace{-22pt}
 \begin{figure}