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h!NeNQ#nsr^)zjJB0*)1djf;(w9gc!RLP-+tzW|7|N{Ijf diff --git a/main.tex b/main.tex index 7c3b80c..1595376 100644 --- a/main.tex +++ b/main.tex @@ -176,6 +176,14 @@ Consider the Lagrangian dual, % At this point, it becomes clear how the normalized min-cut is implicated in \cite{vygen_fptas_2024}. The optimum of normalized min-cut is exactly the value of $\lambda$ when $L(\lambda)$ is 0. +We have shown that the budget $B$ in normalized min-cut does not really matter as long as $B>b$. Note that $L(\lambda)$ and the normalized min-cut look similar to the principal sequence of partitions of a graph and the graph strength problem. + +\subsection{graph strength} +Given a graph $G=(V,E)$ and a cost function $c:V\to \Z_+$, the strength $\sigma(G)$ is defined as $\sigma(G)=\min_{\Pi}\frac{c(\delta(\Pi))}{|\Pi|-1}$, where $\Pi$ is any partition of $V$, $|\Pi|$ is the number of parts in the partition and $\delta(\Pi)$ is the set of edges between parts. + +The principal sequence of partitions of $G$ is a piecewise linear concave curve $L(\lambda)=\min_\Pi c(\delta(\Pi))-\lambda |\Pi|$. + +(there is a $\pm1$ difference...) \section{Random Stuff}