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main.tex

@@ -72,9 +72,18 @@ which shows $c(E)-c(\cl(F_1\cup F_2))\leq c(E)-c(F_1\cup F_2)<\sigma(r(E)-r(\cl(
 \item 
 % I don't think this is true on general matroids.
 We define the ideal base packing $y$ by induction. Suppose that the ideal base packing $y'$ on $M|F^*$ is known.
-Let $y^*$ be the optimal fractional base packing of $M$ with capacity $c$. (Following Thorup's notations, $y^*(B)$ is a probability on the set of bases and ``optimal'' means that $1/\sigma = \max_e \frac{\sum_{B:e\in B}y^*(B)}{c(e)}$.)
-We uniformly and independently choose a base $B_{F^*}$ in $M|F^*$ and a base $B$ in $M$. Note that the set $S=B_{F^*}\cup (B\setminus F)$ is always a base if $M$ is graphic.
-For general matroids the size of $S$ is $r(F)+r(E)-|B\cap F|$.
+Let $y^*$ be the optimal fractional base packing of $M$ with capacity $c$. 
+(Following Thorup's notation, $y^*(B)$ is a probability on the set of bases and ``optimal'' means that $1/\sigma = \max_e \frac{\sum_{B:e\in B}y^*(B)}{c(e)}$.)
+
+We uniformly and independently choose a base $B_{F^*}$ in the support of $y'$ and a base $B$ in the support of $y^*$ and construct a new set $S=B_{F^*}\cup (B\setminus F^*)$.
+For any base $B$, the size of $S$ is $r(F^*)+r(E)-|B\cap F^*|$. However, if $B$ is in the support of $y^*$ then $|S|$ is exactly $r(E)$. To see this, consider the average relative load of $y^*$ on $e\in E\setminus F^*$.
+\[
+\sum_{e\in E\setminus F^*} \frac{c(e)}{c(E\setminus F^*)} \left[\sum_{B:e\in B} \frac{\Pr[B]}{c(e)}\right] 
+\geq \frac{r(E)-r(F^*)}{c(E\setminus F^*)}=\frac{1}{\sigma}
+\]
+Note that $B$ is taken from the support of the optimal base packing, then $\sum_{B:e\in B} \frac{\Pr[B]}{c(e)}$ are the same for all elements and every $B$ contains $r(E)-r(F^*)$ edges in $E\setminus F^*$.
+
+In graphic matroids it follows easily that $S$ is a spanning tree. In general matroids $S$ may not be independent.
 \end{enumerate}
 \end{proof}