z

main.tex

@@ -178,6 +178,9 @@ for any base $B$ of $M'$, the expected size of intersection is at most $O(k')$.
 A counterexample would be uniform matroids $U_{2n,n}$.
 The size of every $k'$-cocycle is $k'+n$, and for any $k'$-cocycle there are bases using $O(n)$ elements in the cocycle.
 
+\paragraph{Nice properties}
+$\cl(F+e)\setminus F$ is a partition of $E\setminus F$ for general matroids.
+
 \subsection{Support size}
 
 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.
@@ -186,6 +189,8 @@ The support size can be exponential. Consider a path with $n$ points and paralle
 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 and $O(m^k)$ bases where $m$ is the number of elements.
 
+Note that LP gives an ideal tree packing with $O(m)$ support size.
+
 \subsection{Rigidity matroids}
 \begin{conjecture}\label{conj:idealrigidbase}
 Let $M$ be a connected 2D rigidity matroid on graph $G=(V,E)$. Let $F^*$ be the optimal flat for strength $F^*=\argmin_{F\subset E}\frac{c(E\setminus F)}{r(E)-r(F)}$.