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main.tex

@@ -149,11 +149,11 @@ However, Lemma~7 in \cite{Thorup_2008} does not generalize to all matroids and w
 
 \subsection{Support size}
 
-Recall that we construct the ideal base packing recursively. Suppose that the ideal base packing for $M|F^*$ is has $n$ bases and let $m$ be the size of support of the optimal base packing of $M$. Then the number of bases in the ideal base packing of $M$ is $nm$. Note that $m$ is upperbounded by the size of the groundset.
+Recall that we construct the ideal base packing recursively. Suppose that the ideal base packing for $M|F^*$ contains $n$ bases and let $m$ be the size of support of the optimal base packing of $M$. Then the number of bases in the ideal base packing of $M$ is $nm$. Note that $m$ is upperbounded by the $|E|$ since the number of constraints is at most $|E|$ in the tree packing LP.
 The support size can be exponential. Consider a path with $n$ points and parallel edges. The depth of recursion can be $n-1$.
 
 Do we need all bases in the packing? Say we are interested in the minimum $k$-cocycle and want to show that we can find a set of bases such that for any minimum $k$-cocycle there is a base whose intersection with the cocyle is at most $O(k)$ elements.
-The strategy is to first find a $\geq k$-cocycle using the utilization algorithm, then randomly delete edges in the $\geq k$-cocycle to make the rank defieciency exactly $k$. Notice that only elements in the $\geq k$-cocycle matter. Thus we only need constant recursion depth.
+The strategy is to first find a $\geq k$-cocycle using the utilization algorithm, then randomly delete edges in the $\geq k$-cocycle to make the rank defieciency exactly $k$. Notice that only elements in the $\geq k$-cocycle matter. Thus we only need constant recursion depth and $O(m^k)$ bases where $m$ is the number of elements.
 
 \subsection{Rigidity matroids}
 \begin{conjecture}\label{conj:idealrigidbase}