From 7f1be75486f60ebefe9b5e0ea81cfa19a0db7221 Mon Sep 17 00:00:00 2001 From: Yu Cong Date: Wed, 3 Dec 2025 12:20:51 +0800 Subject: [PATCH] z --- main.tex | 3 ++- 1 file changed, 2 insertions(+), 1 deletion(-) diff --git a/main.tex b/main.tex index 085f879..eb45389 100644 --- a/main.tex +++ b/main.tex @@ -152,7 +152,8 @@ However, Lemma~7 in \cite{Thorup_2008} does not generalize to all matroids and w 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. 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? +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. \subsection{Rigidity matroids} \begin{conjecture}\label{conj:idealrigidbase}