weight truncation!

main.tex

@@ -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.
 \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}
     \autoref{lp:cutinterdict} has an integrality gap of 2.
 \end{conjecture}