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main.tex

@@ -128,24 +128,19 @@ Ideal base packing is a distribution on some bases. Given a subset $D\subset E$,
 The expectation is exactly $\sum_{e\in D} c(e)u^*(e)$.
 
 Recall that our goal is to show that for any minimum $k$-cocycle a random base in the ideal base packing uses $O(k)$ elements in the cocycle in expectation.
-Thorup proves that there is a special $k$-cocycle that for any 
+Thorup proves in \textbf{graphic matroids} that there is a fixed distribution of $k$-cocycles such that for any base from the ideal base packing, the expected size of intersection is at most $O(k)$ (Lemma~7 in \cite{Thorup_2008}).
+Then it follows that if we take a random spanning tree from the ideal tree packing, there is a fixed $(k+1)$-cut $C$ such that the expected size of intersection is at most $O(k)$, which implies $\sum_{e\in C}c(e)u^*(e)\in O(k)$.
 
-Lemma~7 in \cite{Thorup_2008} does not generalize to all matroids.
+How is this fixed $(k+1)$-cut (or $k$-cocycle) related to the minimum $(k+1)$-cut (minimum $k$-cocycle) ?
+The minimum $k$-cocycle has smaller capacity than $C$.
+Note that Thorup uses a greedy way to construct the cocycle $C$. Elements in $C$ always has the largest possible utilization.
+These facts implies that the minimum $k$-cocycle has a smaller value $\sum c(e)u^*(e)$ than $C$.
 
-Let $F^*$ be the optimal flat.
-Choose an random element $f\in E-F^*$ and construct a new flat $F=\cl(F^*+f)$ and repeat this process until $r(F)=r(E)-k$. Let $X$ be a independent set with rank $r(E)-r(F^*)$ inside the cocycle $E-F^*$.
-We generate another flat $F\supset F^*$ in the following way. 
-Initially we set $F=F^*$.
-Randomly choose an edge $f$ in $E-F$, update $F$ to $\cl(F+f)$. Repeat this operation $t$ times for a fixed $t< r(E)-r(F^*)$.
-We want to upperbound the expected size of $X\setminus F$ using $c(r(E)-r(F))$.
-\note{This seems stronger than constant gap...}
-
-In graphs this seems like a random contraction on $G/\mathcal F^*$. However, instead of the probability certain mincut is preserved, we are interested the expected number of remaining edges of a spanning tree.
+However, Lemma~7 in \cite{Thorup_2008} does not generalize to all matroids and we need to dive into the construction of $C$.
 
 \subsection{Support size}
-...
 
-\subsection*{Rigidity matroids}
+\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 $F^*=\argmin_{F\subset E}\frac{c(E\setminus F)}{r(E)-r(F)}$.
 Let $X\subset E\setminus F^*$ be a independent set with rank $r(E)-r(F^*)$. Then for any maximal independent set $B_{F^*}\subset F^*$, $X\cup B_{F^*}$ is a base of $M$.
@@ -157,14 +152,6 @@ For a subset of rigid components $\mathcal P$, let $t=|\bigcup_{P\in \mathcal P}
 One can see that in this process we do not care the actual base $B_{F^*}$ and only the 1-thin cover matters.
 \end{remark}
 
-\begin{conjecture}
-Assume \autoref{conj:idealrigidbase} is true.
-Then there is a hyperplane $H$ such that  $|X\setminus H|\leq c$ for some constant $c$.
-\end{conjecture}
-
-If what we need is only ``existance'' of such a hyperplane, sure there is one.
-Need to really understand ideal tree packing. How to find a tree that is hit by certain minimum cocircuit constant times?
-
 \section{Greedy base packing}