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zNKj&gDPJ!8c1@49OETH3rau5Wj~u_-=hk1i>WDP8O{y0eM#RZ9;m27X(Zn^eidJJ% zAHvS5XBh6rel>2Q=C}lU_HZwi4^7`^P+{`1S&lMzS}!tgpOi}t{H-d@3hi*|Ib5W?q&DpwHEZG~%Ds@MwH5VN?*WK6SQu0sp{N%jOCuRDi zxOO7GZcBa|pW5jr>;9B`%X*}@jbQRLxjIgI=pjpDr?8VS#9!^F4% diff --git a/main.tex b/main.tex index 0e9c6b3..17d46c3 100644 --- a/main.tex +++ b/main.tex @@ -268,13 +268,23 @@ The proof is using the argument for showing $\lambda_1=\min \frac{w(C\setminus F \autoref{lp:conninterdict} has an integrality gap of 4. \end{conjecture} -\paragraph{The plan} Assume the optimal $\lambda$ in \autoref{lp:dualcutint} is indeed $\frac{w(C\setminus F)}{b+k-c(F)}$, \autoref{conj:intgap} implies a stronger version of \autoref{thm:main}. Let $\lambda^*$ be the optimal solution to interdiction LP\ref{lp:dualcutint}. Let $w_{\lambda^*}$ be the truncated weight. Denote the fractional and integral solution to the mincut problem(tree packing) on weight $w_{\lambda^*}$ by $\Lambda_{w_{\lambda^*}}^{fr}$ and $\Lambda_{w_{\lambda^*}}^{int}$(subscripts will be ignored). We have $2\Lambda^{fr}\geq \Lambda^{int}$ by the integrality gap of graph cut. Let $(C^*,F^*)$ be the optimal solution to \autoref{lp:conninterdict} and let $\opt(IP)$ be the optimum. Recall that we have $w_{\lambda^*}(C^*)\leq b\lambda^*+\opt(IP)$. \autoref{conj:intgap} implies that -\begin{align*} -4\opt(LP) &=4\Lambda^{fr}-4b\lambda^*\\ - &\geq 2\Lambda^{int} - 4b\lambda^*\\ - &\geq w_{\lambda^*}(C^*)-b\lambda^*. -\end{align*} -The last inequality shows \autoref{thm:main}. +However, \autoref{conj:intgap} is wrong. The integrality gap is unbounded. 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. + +What is the gap if we only relax $\lambda$ in the Lagrangian dual? \section{Random Stuff}