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main.tex

@@ -172,16 +172,16 @@ For general matroids, we want to show the following.
 \begin{conjecture}\label{conj:dist}
 Let $M'$ be the contraction minor $M/F^*$. The rank of $M'$ is $k$. 
 Given a positive integer $k'<k$, then there exists a distribution on $k'$-cocycles such that
-for any base $B$ of $M'$, the expected size of intersection is at most $O(k')$.
+for any base $B$ of $M'$, the expected size of intersection is $O(1)$ for fixed $k'$.
 \end{conjecture}
 
 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.
 
-\begin{proposition}
+\begin{lemma}\label{lem:partition}
 Let $M=(E,\mathcal I)$ be a matroids and let $F$ be a flat of $M$.
 Then $\cl(F+e)\setminus F$ is a partition of $E\setminus F$.
-\end{proposition}
+\end{lemma}
 \begin{proof}
 Suppose for contradiction that there are two elements $x,y\in E-F$ such that $\cl(F+x)\setminus F$ and $\cl(F+y)\setminus F$ have non-empty intersection. Let $z$ be an element in the intersection.
 Then we find a circuit $C_{x,z}$ in $\cl(F+e)$ such that $C_{x,z}\setminus F= \set{x,z}$.
@@ -216,6 +216,14 @@ One can see that in this process we do not care the actual base $B_{F^*}$ and on
 \autoref{conj:dist} is true when $M$ is a 2D rigidity matroid.
 \end{conjecture}
 
+Try to minic Thorup's proof for graphic matroids. As \autoref{lem:partition} states, spans form a partition on $E-F$.
+Note that for graphic matroids the number of spans is $O((r(E)-r(F))^2)$.
+For any spanning tree $T$, the number of spans hitting $T$ is exactly $r(E)-r(F)$ and these spans have a nice structure. If we contract each component in $G[F]$ to a vertex and consider spans a set of parallel edges, then the set of spans hitting $T$ is a tree (with parallel edges) in $G[F]$.
+For rigidity matroids, the number of rigid components in $F$ cannot be bounded by $r(E)-r(F)$.\footnote{Since the 1-thin cover inducing a cocircuit can have any number of rigid components.}
+
+Let $\mathcal S_F$ be the span partition of $E-F$ described in \autoref{lem:partition}. Let $B$ be a fixed base. We say a part $S\in \mathcal S_F$ is good if $S\cap B$ is non-empty. Note that if we merge a part $S$ into $F$, then the resulting span partition is a coarsening of $\mathcal S_F$. For general matroids, the set of good parts in $\mathcal S_F$ never merge when merging a good part into $F$.
+It would be nice if we can characterize good parts in rigidity matroids with 1-thin cover.
+
 \section{Greedy base packing}