more on strength

main.tex

@@ -182,10 +182,14 @@ We have shown that the budget $B$ in normalized min-cut does not really matter a
 Assume that the graph $G$ is connected (otherwise add dummy edges). 
 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. Note that an alternative formulation of strength (using graphic matroid rank) is $\sigma(G)=\min_{F\subset E} \frac{|E-F|}{r(E)-r(F)}$, which in general is the fractional optimum of matroid base packing.
 
-The principal sequence of partitions of $G$ is a piecewise linear concave curve $L(\lambda)=\min_\Pi c(\delta(\Pi))-\lambda |\Pi|$. Cunningham used principal partition to computed graph strength\cite{cunningham_optimal_1985}. There is a list of good properties mentioned in \cite[Section 6]{chekuri_lp_2020}.
+The principal sequence of partitions of $G$ is a piecewise linear concave curve $L(\lambda)=\min_\Pi c(\delta(\Pi))-\lambda |\Pi|$. Cunningham used principal partition to computed graph strength\cite{cunningham_optimal_1985}. There is a list of good properties mentioned in \cite[Section 6]{chekuri_lp_2020}(implicated stated in \cite{cunningham_optimal_1985}).
 \begin{itemize}
 \item $L(\lambda)$ is piecewise linear concave since it is the lower envelope of some line arrangement.
-    \item Consider two adjacent breakpoints on $L$...
+\item For each line segment on $L(\lambda)$ there is a corresponding partition $\Pi$. If $\lambda^*$ is a breakpoint on $L(\lambda)$, then there are two optimal solution (say partitions $P_1$ and $P_2$, assuming $|P_1|\leq|P_2|$) to $\min_\Pi c(\delta(\Pi))-\lambda^* |\Pi|$.  Then $P_2$ is a refinement of $P_1$.
+    \begin{proof}[sketch]
+        Suppose that $P_2$ is not a refinement of $P_1$. We claim that the meet of $P_1$ and $P_2$ achieves a smaller objective value than $P_1$ or $P_2$ does. For simplicity we assume $G$ is connected. The correspondence between graphic matroid rank function and partitions of $V$ gives us a reformulation $L(\lambda^*)=\min_{F\subset E}c(E-F)-\lambda^*(r(E)-r(F)+1)$. Then the claim is equivalent to the fact that for two optimal solutions $F_1,F_2$ to $L(\lambda^*)$, $F_1\subset F_2$, which can be seen by submodularity of matroid rank functions.
+    \end{proof}
+\item Let $\lambda^*$ be a breakpoint on $L(\lambda)$ induced by edge set $F$. The next breakpoint is induced by the edge set $F'$ such that $F'$ contains $F$ and $F'-F$ is the solution to strength problem on the smallest strength component of $G\setminus F$.
 \end{itemize}
 
 (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 $\lambda$)