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Yu Cong
2025-12-04 11:48:23 +08:00
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@@ -160,9 +160,24 @@ One can carefully design the distributions for contractions so instead of contra
We uniformly and randomly choose $k-k'+1$ parts in $\mathcal P_F$ and merge them into a big part. We uniformly and randomly choose $k-k'+1$ parts in $\mathcal P_F$ and merge them into a big part.
Denote the resulting partition by $\mathcal P_{F'}$. Denote the resulting partition by $\mathcal P_{F'}$.
Let $T$ be a spanning tree in the support of ideal tree packing. Let $T$ be a spanning tree in the support of ideal tree packing.
Note that the number of inter-component edges of $T$ in $\mathcal P_F$ is $k$. Recall that the number of inter-component edges of $T$ in $\mathcal P_F$ is $k$.
Then we have
\[
\E_{F'}[|T\setminus F'|]=k\brack{1-\paren{\frac{k+1-k'}{k+1}}^2}\leq 2k'.
\]
\end{proof} \end{proof}
For general matroids, we want to show the following.
\begin{conjecture}
Let $M'$ be the contraction $M/F^*$. 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 $O(k')$.
\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.
\subsection{Support size} \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. 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.