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add a lemma

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Yu Cong 2025-05-05 15:00:38 +08:00
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@ -147,9 +147,12 @@ What is the first breakpoint on $L(\mu)$?
We have $-\mu(b-\lambda_c)\leq w(C- F)-\mu(b-c(F))$ for any cut $C$ and $F\subsetneq C$. Thus the first breakpoint is $\mu=\min \frac{w(C- F)}{\lambda_c-c(F)}$, which is the value of normalized mincut. We have $-\mu(b-\lambda_c)\leq w(C- F)-\mu(b-c(F))$ for any cut $C$ and $F\subsetneq C$. Thus the first breakpoint is $\mu=\min \frac{w(C- F)}{\lambda_c-c(F)}$, which is the value of normalized mincut.
\newline \newline
What about other breakpoints? Currently I can only prove the following :( What about other breakpoints?
\begin{lemma} \begin{lemma}
$\lambda_i=\min \frac{w(C- F)-w(C_{i-1}- F_{i-1})}{c(F_{i-1})-c(F)}$, where the minimum is taken over all cut $C$ and $F\subset C$ such that both the numerator and denominator are positive. $\mu^*\in [\min\limits_{C,F} \frac{w(C\setminus F)}{\lambda_c - c(F)},\min\limits_{C,F} \frac{w(C\setminus F)}{b - c(F)}]$
\end{lemma}
\begin{lemma}
$\mu_i=\min \frac{w(C\setminus F)-w(C_{i-1}\setminus F_{i-1})}{c(F_{i-1})-c(F)}$, where the minimum is taken over all cut $C$ and $F\subset C$ such that both the numerator and denominator are positive.
\end{lemma} \end{lemma}
\end{frame} \end{frame}