better pf sketch

main.tex

@@ -187,7 +187,7 @@ The principal sequence of partitions of $G$ is a piecewise linear concave curve
 \item $L(\lambda)$ is piecewise linear concave since it is the lower envelope of some line arrangement.
 \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.
+        Suppose that $P_2$ is not a refinement of $P_1$. We claim that the meet of $P_1$ and $P_2$ achieves a objective value at least no larger 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 $L(\lambda^*)=\min_{F\subset E}c(E-F)-\lambda^*(r(E)-r(F)+1)$. Let $g(F)=c(E-F)+\lambda^*r(F)-\lambda^* n$. Then the claim is equivalent to the fact that for two optimal solutions $F_1,F_2$ to $L(\lambda^*)$, $g(F_1\cup F_2)\leq g(F_1)=g(F_2)\leq g(F_1\cap F_2)$, 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}