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main.tex

@@ -16,9 +16,13 @@
 
 \section{Ideal base packing}
 Try to generalize Thorup's ideal tree packing \cite{Thorup_2008} to matroids. 
-Certainly it won't work on all matroids.
+We cannot expect it to work on all matroids.
 The goal is to figure out some sufficient conditions and their relations with basepacking($\lambda\leq c \sigma$) and random contraction($\lambda \leq c \frac{|E|}{r(E)}$).
 
+The idea is that we want to find a small set of bases such that for any minimum $k$-cocycle there is a base that the cocycle hits the base $O(k)$ times.
+The number of bases should be as small as possible.
+One can select all bases and see that for every $k$-cocycle there is a base that only gets hit exactly $k$ times.
+
 Let $M=(E,\mathcal I)$ be a matroid with capacity $c:E\to \R_{\geq 0}$ on elements and let $\sigma=\min_{F\subset E} \frac{c(E-F)}{r(E)-r(F)}$ be its weighted strength.
 
 \begin{algo}
@@ -90,6 +94,8 @@ Then the lemma follows by induction.
 \end{enumerate}
 \end{proof}
 
+The following lemma does not seem related to the ideal tree packing.
+
 \begin{lemma}
 If we decrease the capacity of an edge, no ideal edge utilization decreases.
 \end{lemma}
@@ -116,7 +122,14 @@ If we increase the capacity, no ideal edge utilization increases. The proof is s
 Removing (contracting) edges has the same effect on ideal utilization as setting the capacity to $0$ ($\infty$).
 \end{remark}
 
-\subsection{Hard part}
+\subsection{Counting}
+
+Ideal base packing is a distribution on some bases. Given a subset $D\subset E$, consider the expected size of intersection with a base sampled from the ideal distribution. 
+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 
+
 Lemma~7 in \cite{Thorup_2008} does not generalize to all matroids.
 
 Let $F^*$ be the optimal flat.
@@ -129,7 +142,10 @@ We want to upperbound the expected size of $X\setminus F$ using $c(r(E)-r(F))$.
 
 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.
 
-\subsection{Rigidity matroids}
+\subsection{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 $F^*=\argmin_{F\subset E}\frac{c(E\setminus F)}{r(E)-r(F)}$.
 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 $M$.
@@ -143,9 +159,12 @@ One can see that in this process we do not care the actual base $B_{F^*}$ and on
 
 \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}