From 00125786d4da4e683f205fa8f6c1f73c33b4343e Mon Sep 17 00:00:00 2001 From: Yu Cong Date: Tue, 2 Dec 2025 00:05:16 +0800 Subject: [PATCH] z --- main.tex | 29 ++++++++--------------------- 1 file changed, 8 insertions(+), 21 deletions(-) diff --git a/main.tex b/main.tex index db20452..e847556 100644 --- a/main.tex +++ b/main.tex @@ -128,24 +128,19 @@ Ideal base packing is a distribution on some bases. Given a subset $D\subset E$, 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 +Thorup proves in \textbf{graphic matroids} that there is a fixed distribution of $k$-cocycles such that for any base from the ideal base packing, the expected size of intersection is at most $O(k)$ (Lemma~7 in \cite{Thorup_2008}). +Then it follows that if we take a random spanning tree from the ideal tree packing, there is a fixed $(k+1)$-cut $C$ such that the expected size of intersection is at most $O(k)$, which implies $\sum_{e\in C}c(e)u^*(e)\in O(k)$. -Lemma~7 in \cite{Thorup_2008} does not generalize to all matroids. +How is this fixed $(k+1)$-cut (or $k$-cocycle) related to the minimum $(k+1)$-cut (minimum $k$-cocycle) ? +The minimum $k$-cocycle has smaller capacity than $C$. +Note that Thorup uses a greedy way to construct the cocycle $C$. Elements in $C$ always has the largest possible utilization. +These facts implies that the minimum $k$-cocycle has a smaller value $\sum c(e)u^*(e)$ than $C$. -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 generate another flat $F\supset F^*$ in the following way. -Initially we set $F=F^*$. -Randomly choose an edge $f$ in $E-F$, update $F$ to $\cl(F+f)$. Repeat this operation $t$ times for a fixed $t< r(E)-r(F^*)$. -We want to upperbound the expected size of $X\setminus F$ using $c(r(E)-r(F))$. -\note{This seems stronger than constant gap...} - -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. +However, Lemma~7 in \cite{Thorup_2008} does not generalize to all matroids and we need to dive into the construction of $C$. \subsection{Support size} -... -\subsection*{Rigidity matroids} +\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 $M$. @@ -157,14 +152,6 @@ For a subset of rigid components $\mathcal P$, let $t=|\bigcup_{P\in \mathcal P} 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} -\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}