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main.tex

@@ -116,6 +116,11 @@ 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}
+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 want to upperbound the expected size of $X\setminus F$ using $c(r(E)-r(F))$.
+
 \subsection{Rigidity matroids}
 \begin{conjecture}
 Let $M$ be a connected 2D rigidity matroid on graph $G=(V,E)$. Let $F^*$ be the optimal flat for strength $F^*=\argmin_{F\subset E}\frac{c(E\setminus F)}{r(E)-r(F)}$.
@@ -124,7 +129,7 @@ Let $X\subset E\setminus F^*$ be a independent set with rank $r(E)-r(F^*)$. Then
 \begin{remark}
 The intuition is that rigidity of $F^*\cup X$ only depends on the 1-thin cover of $F^*$ but not the base $B_{F^*}$.
 Consider a non-proper 1-thin cover where the rigid components come from those of 1-thin cover of $F^*$ and singleton elements of $X$. A proper 1-thin cover can be computed through coarsening.
-For a subset of rigid components $\mathcal P$, let $t=|\bigcup_{P\in \mathcal P} V[P]|$ be the number of vertices. If the number of edges $|\bigcup_{P\in \mathcal P} P|$ is at least $2t-3$ then we merge these components into a new one.
+For a subset of rigid components $\mathcal P$, let $t=|\bigcup_{P\in \mathcal P} V[P]|$ be the number of vertices. If the number of edges $\sum_{P\in \mathcal P} 2|P|-3$ is at least $2t-3$ then we merge these components into a new one.
 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}