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main.tex

@@ -84,6 +84,9 @@ For any base $B$, the size of $S$ is $r(F^*)+r(E)-|B\cap F^*|$. However, if $B$
 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.
+\note{From now on we assume $S$ is a base. This should holds in all $(k,2k-1)$-sparsity matroids.}
+Then the lemma follows by induction.
+\item If $F^*=\emptyset$ or the size of the groundset is 1, then one can easily see the claim holds since every element have the same ideal utilitization. Now suppose the claim holds for $M|F^*$. In the preceeding bullet point we have already shown that every element in $E\setminus F^*$ have the same utilization. Note that we also have shown in the first bullet point that the ideal utilization is larger in $M$ than in $M|F^*$. Thus, the construction conincides with the greedy algorithm for minimum matroid base.
 \end{enumerate}
 \end{proof}