From ed60c4820f5b8e95eb4ab2f2f75fb3e4979d2829 Mon Sep 17 00:00:00 2001 From: Yu Cong Date: Tue, 25 Nov 2025 17:17:53 +0800 Subject: [PATCH] z --- main.tex | 7 ++++++- 1 file changed, 6 insertions(+), 1 deletion(-) diff --git a/main.tex b/main.tex index fdbcff2..4ac13d9 100644 --- a/main.tex +++ b/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}