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main.tex

@@ -71,7 +71,10 @@ 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 fractional base packing $y$ recursively. Suppose that we have the desired fraction packing $y'$ on $M|F^*$...
+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 $r(F)+r(E)-|B\cap F|$.
 \end{enumerate}
 \end{proof}