update proof for graphic matroid

.zed/tasks.json

@@ -1,25 +0,0 @@
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main.tex

@@ -150,7 +150,7 @@ Let $F$ be the optimal flat for strength and assume $k=r(E)-r(F)>k'$. We want to
 Basically we need to do random contraction on $M/ F$. Let $\mathcal X$ be the set $\set{X|X=B\setminus F \land r(X)=k}$. That is, we consider all bases that hitten by the $k$-cocycle exactly $k$ times and for each of them we collect the intersection with the $k$-cocycle.
 Then we do $k-k'$ random contractions in $M/F$ to get a random $k'$-cocycle $C_{k'}$.
 
-\begin{lemma}[Lemma~7 in \cite{Thorup_2008}, restated]
+\begin{lemma}[Lemma~7 in \cite{Thorup_2008}, restated]\label{lem:idealload}
 	For graphic matroids, there is a distribution on $C_{k'}$ such that for any base in the ideal base packing, the expected size of its intersection with $C_{k'}$ is at most $2k'$.
 \end{lemma}
 \begin{proof}
@@ -167,16 +167,21 @@ Then we have
 	\]
 \end{proof}
 
+Finally, we want to upperbound the ideal load of the minimum $k$-cocycle ($(k+1)$-cut).
+By \autoref{lem:idealload}, there is a distribution of some $k$-cocycles that use at most $2k$ edges in any ideal spanning tree.
+Then there is a special $k$-cocycle $C$ that use (expectedly) at most $2k$ edges in a random ideal spanning tree.
+Note that for edge set $D$ the ideal load $\sum_{e\in D} u^*(e)c(e)$ can be interpreted as the expected number of edges of $D$ in a random ideal spanning tree.
+Now we consider the minimum $k$-cocycle $\mathcal K$. Its ideal load $\sum_{e\in \mathcal K} u^*(e)c(e)$ must be at most that of $C$, since $C$ has the largest the ideal load per edge capacity and $c(\mathcal K)\leq c(C)$.
+
 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
+	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 $O(1)$ for fixed $k'$.
-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.
+The size of every $k'$-cocycle is $k'+n$, and the size of expected intersection should be $O(n)$.
 
 \begin{lemma}\label{lem:partition}
 	Let $M=(E,\mathcal I)$ be a matroids and let $F$ be a flat of $M$.

ref.bib

@@ -44,7 +44,7 @@
 	language      = {en}
 }
 @article{thorup_fully-dynamic_2007,
-	title         = {Fully-{Dynamic} {Min}-{Cut}\textasteriskcentered},
+	title         = {Fully-{Dynamic} {Min}-{Cut}},
 	volume        = {27},
 	issn          = {0209-9683, 1439-6912},
 	url           = {http://link.springer.com/10.1007/s00493-007-0045-2},