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main.tex

@@ -3,7 +3,7 @@
 \usepackage{algo}
 \geometry{margin=2cm}
 
-\title{Notes on ideal base packing}
+\title{Ideal and greedy base packing}
 \author{}
 \date{}
 
@@ -12,7 +12,7 @@
 \DeclareMathOperator*{\cl}{span}
 
 \begin{document}
-% \maketitle
+\maketitle
 
 \section{Ideal base packing}
 Try to generalize Thorup's ideal tree packing \cite{Thorup_2008} to matroids. 
@@ -117,12 +117,20 @@ Removing (contracting) edges has the same effect on ideal utilization as setting
 \end{remark}
 
 \subsection{Hard part}
+Lemma~7 in \cite{Thorup_2008} does not generalize to all matroids.
+
 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.
 
 \subsection{Rigidity matroids}
-\begin{conjecture}
+\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 $M$.
 \end{conjecture}
@@ -133,25 +141,13 @@ 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{comment}     % principal sequence of partition
-\section{Principal sequence of partition on graphs}
-For a graph $G=(V,E)$ with edge capacity $c:V\to \Z_+$, the strength $\sigma(G)$ is defined as $\sigma(G)=\min_{\Pi}\frac{c(\delta(\Pi))}{|\Pi|-1}$, where $\Pi$ is any partition of $V$, $|\Pi|$ is the number of parts in the partition and $\delta(\Pi)$ is the set of edges between parts. Note that an alternative formulation of strength (using graphic matroid rank function) is $\sigma(G)=\min_{F\subset E} \frac{c(E-F)}{r(E)-r(F)}$, which in general is the fractional optimum of matroid base packing.
+\begin{conjecture}
+Assume \autoref{conj:idealrigidbase} is true.
+???
+\end{conjecture}
+
+\section{Greedy base packing}
 
-The principal sequence of partitions of $G$ is a pwl concave curve $L(\lambda)= \min_\Pi c(\delta(\Pi))-\lambda |\Pi|$. (alternatively, $L(\lambda)=\min_{F\in E}c(E\setminus F)-\lambda(r(E)-r(F)+1)$) Cunningham used principal partition to computed graph strength\cite{cunningham_optimal_1985}. There is a list of good properties mentioned in \cite[Section 6]{chekuri_lp_2020}(implicated stated in \cite{cunningham_optimal_1985}).
-\begin{enumerate}
-\item We can assume $G$ is connected and deal with the smallest strength component. One can see this by fractional base packing on the direct sum of matroids. Note that on disconnected graphs we should use the edge set definition instead of partitions.
-\item $L(\lambda)$ is piecewise linear concave since it is the lower envelope of some line arrangement.
-\item For each line segment on $L(\lambda)$ there is a corresponding partition $\Pi$. If $\lambda^*$ is a breakpoint on $L(\lambda)$, then there are two optimal solution (say partitions $P_1$ and $P_2$, assume $|P_1|\leq|P_2|$) to $\min_\Pi c(\delta(\Pi))-\lambda^* |\Pi|$.  Then $P_2$ is a refinement of $P_1$.
-    \begin{proof}[sketch]
-        Suppose that $P_2$ is not a refinement of $P_1$. We claim that the meet of $P_1$ and $P_2$ achieves a objective value at least no larger than $P_1$ or $P_2$ does. The correspondence between graphic matroid rank function and partitions of $V$ gives us a reformulation $L(\lambda^*)=\min_{F\subset E}c(E-F)-\lambda^*(r(E)-r(F)+1)$. Here $F$ is the set of edges in each part of $\Pi$. Let $g(F)=c(E-F)+\lambda^*r(F)-\lambda^* n$. Then the claim is equivalent to the fact that for two optimal solutions $F_1,F_2$ to $L(\lambda^*)$, $g(F_1\cap F_2)\leq g(F_1)=g(F_2)\leq g(F_1\cup F_2)$, which can be seen by the submodularity of $g$ and the optimality of $F_1,F_2$.
-    \end{proof}
-The number of breakpoints on $L(\lambda)$ is at most $n-1$.
-\item Let $\lambda^*$ be a breakpoint on $L(\lambda)$ induced by edge set $F$. The next breakpoint is induced by the edge set $F'\subset F$ and $F'$ is the solution to strength problem on the smallest strength component of $F$. $\lambda^*$ is the strength of the smallest strength component in $F$. These claims can be seen by the following arguments. From the previous bullet we have $\min_{\Delta F} c(E-F+\Delta F)-\lambda^*(r(E)-r(F-\Delta F)+1)=L(\lambda^*)$. Consider the largest $\lambda^*$ which allows $\Delta F=\emptyset$ to be an optimal solution. Such $\lambda^*$ would be the next breakpoint. For any $\Delta F$, $c(E-F+\Delta F)-\lambda^*(r(E)-r(F-\Delta F)+1)\geq c(E-F)-\lambda^*(r(E)-r(F)+1)$. Thus we have $\lambda^*\leq \frac{c(\Delta F)}{r(F)-r(F-\Delta F)}$.
-\item Consider $\lambda\in [0,\e]$ for some small enough $\e$. The Lagrangian dual $\min_F c(E\setminus F)-\lambda (r(E)-r(F)+1)$ gets the optimum at $F=E$. 
-That is $c(E\setminus F')-\lambda(r(E)-r(F')+1)>-\lambda$ for all $F'\subsetneq E$.
-We are interested in the upperbound $\e$ of $\lambda$ such that the optimal $F$ is a proper subset of $E$ when $\lambda >\e$. Therefore, the upperbound is $\e=\min_{F\subsetneq E}\frac{c(E\setminus F)}{r(E)-r(F)}$, which is exactly the strength.
-\end{enumerate}
-\end{comment}
 
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