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weight truncation!

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Yu Cong 2025-04-21 15:46:15 +08:00
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@ -248,6 +248,14 @@ We want to prove something like tree packing theorem for \autoref{lp:dualcutint}
The optimum of \autoref{lp:dualcutint} is $\min \set{\frac{w(C\setminus F)}{B-c(F)}| \forall \text{cut $C$}, c(F)\leq b}$, where $B$ is the cost of mincut in $G$ and $b$ is the budget. The optimum of \autoref{lp:dualcutint} is $\min \set{\frac{w(C\setminus F)}{B-c(F)}| \forall \text{cut $C$}, c(F)\leq b}$, where $B$ is the cost of mincut in $G$ and $b$ is the budget.
\end{conjecture} \end{conjecture}
I believe the previous conjecture is not likely to be true. This one seems better,
\begin{conjecture}\label{conj:optimaldual}
$\lambda=\min \frac{w(C\setminus F)}{B-c(F)}$ is optimal for \autoref{lp:dualcutint}.
\end{conjecture}
Assuming \autoref{conj:optimaldual} is true, \autoref{lp:dualcutint} in fact gives the idea of weight truncation. The capacity of each edge $e$ in the ``tree packing'' is $\min\{c(e)\lambda,w(e)\}$.
\begin{conjecture} \begin{conjecture}
\autoref{lp:cutinterdict} has an integrality gap of 2. \autoref{lp:cutinterdict} has an integrality gap of 2.
\end{conjecture} \end{conjecture}