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Yu Cong 2025-04-19 00:40:03 +08:00
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@ -169,7 +169,7 @@ Note that now this is almost a positive covering LP. Let $L(\lambda)= \min \{ w(
\}$. \}$.
Consider the Lagrangian dual, Consider the Lagrangian dual,
\begin{equation*} \begin{equation*}
\max_{\lambda\geq 0} L(\lambda)= \max_{\lambda\geq 0} \min \left\{ w(C\setminus F)-\lambda(b-c(F)), \forall \text{cut $C$}\;\forall F\subset C \max_{\lambda\geq 0} L(\lambda)= \max_{\lambda\geq 0} \min \left\{ w(C\setminus F)-\lambda(b-c(F)) | \forall \text{cut $C$}\;\forall F\subset C
% \land c(F)\leq b % \land c(F)\leq b
\right\} \right\}
\end{equation*} \end{equation*}
@ -197,7 +197,7 @@ The number of breakpoints on $L(\lambda)$ is at most $n-1$.
(there is a $\pm1$ difference between principal partition and graph strength... but we dont care those $c\lambda$ terms since the difficult part is minimize $L(\lambda)$ for fixed $\lambda$) (there is a $\pm1$ difference between principal partition and graph strength... but we dont care those $c\lambda$ terms since the difficult part is minimize $L(\lambda)$ for fixed $\lambda$)
\subsection{principal sequence of partitions for cut interdiction} \subsection{principal sequence of partitions for cut interdiction}
I don't expect similar results hold. Now we focus One $L(\lambda)=\min \{w(C\setminus F)-\lambda(b-c(F)) | \forall \text{cut } C\;\forall F\subset C\}$. We can still assume that $G$ is connected and see that $L(\lambda)$ is pwl concave. Let $\lambda^*$ be a breakpoint on $L$. Suppose that there are two optimal solutions $(C_1,F_1)$ and $(C_2,F_2)$ at $\lambda^*$. For fixed $C$ ($C_1=C_2$), the same argument for principal partition still works. However, the difficult part is that $C$ might not be the same.
\subsection{integrality gap} \subsection{integrality gap}
I guess the 2 approximation cut enumeration algorithm implies a integrality gap of 2 for cut interdiction problem. I guess the 2 approximation cut enumeration algorithm implies a integrality gap of 2 for cut interdiction problem.