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main.tex

@@ -146,9 +146,22 @@ Note that Thorup used a greedy way to construct the cocycle $C$. Elements in $C$
 These facts implies that the minimum $k$-cocycle has a smaller value $\sum c(e)u^*(e)$ than $C$.
 
 However, Lemma~7 in \cite{Thorup_2008} does not generalize to all matroids and we need to take a close look at the construction of $C$.
-Let $k'$ be $r(E)-r(F^*)$ and we want to find the minimum $k$-cocycle.
-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.
-...
+Let $F$ be the optimal flat for strength and assume $k=r(E)-r(F)>k'$. We want to find the minimum $k'$-cocycle.
+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]
+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}
+We can assume the matroid is connected. (Otherwise we can remove loops and coloops and add dummy elements.)
+In graphic matroids, $F$ corresponds to a partition $\mathcal P_F$ with $k+1$ parts, where each part is the vertex set of a component in $G[F]$.
+One can carefully design the distributions for contractions so instead of contracting edges, we consider randomly merging parts in $\mathcal P_F$.
+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 $\mathcal P_{F'}$.
+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 $k$.
+\end{proof}
 
 \subsection{Support size}