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Yu Cong
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main.tex
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@@ -15,7 +15,7 @@
% \maketitle % \maketitle
\section{Ideal base packing} \section{Ideal base packing}
Try to generalize Thorup's ideal tree packing \cite{Thorup2008} to matroids. Try to generalize Thorup's ideal tree packing \cite{Thorup_2008} to matroids.
Certainly it won't work on all matroids. Certainly it won't work on all matroids.
The goal is to figure out some sufficient conditions and their relations with basepacking($\lambda\leq c \sigma$) and random contraction($\lambda \leq c \frac{|E|}{r(E)}$). The goal is to figure out some sufficient conditions and their relations with basepacking($\lambda\leq c \sigma$) and random contraction($\lambda \leq c \frac{|E|}{r(E)}$).
@@ -123,6 +123,9 @@ Let $X\subset E\setminus F^*$ be a independent set with rank $r(E)-r(F^*)$. Then
\end{conjecture} \end{conjecture}
\begin{remark} \begin{remark}
The intuition is that rigidity of $F^*\cup X$ only depends on the 1-thin cover of $F^*$ but not the base $B_{F^*}$. 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.
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} \end{remark}
\begin{comment} % principal sequence of partition \begin{comment} % principal sequence of partition

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ref.bib
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@@ -32,17 +32,20 @@
pages = {1334--1353}, pages = {1334--1353},
} }
@article{Thorup2008, @inproceedings{Thorup_2008,
address = {Victoria British Columbia Canada},
title = {Minimum k-way cuts via deterministic greedy tree packing}, title = {Minimum k-way cuts via deterministic greedy tree packing},
issn = {07378017}, ISBN = {9781605580470},
url = {http://dl.acm.org/citation.cfm?id=1374402}, url = {https://dl.acm.org/doi/10.1145/1374376.1374402},
doi = {10.1145/1374376.1374402}, DOI = {10.1145/1374376.1374402},
journal = {Proceedings of the 40th annual ACM symposium on …}, booktitle = {Proceedings of the fortieth annual ACM symposium on Theory of
computing},
publisher = {ACM},
author = {Thorup, Mikkel}, author = {Thorup, Mikkel},
year = {2008}, year = {2008},
note = {ISBN: 9781605580470}, month = may,
keywords = {k-way cuts, tree packing}, pages = {159–166},
pages = {159--165}, language = {en},
} }
@article{thorup_fully-dynamic_2007, @article{thorup_fully-dynamic_2007,
@@ -59,5 +62,7 @@
month = feb, month = feb,
year = {2007}, year = {2007},
pages = {91--127}, pages = {91--127},
file = {Thorup_2007_Fully-Dynamic Min-Cut.pdf:/Users/congyu/Zotero/storage/Q329VHH3/Thorup_2007_Fully-Dynamic Min-Cut.pdf:application/pdf}, file = {Thorup_2007_Fully-Dynamic
Min-Cut.pdf:/Users/congyu/Zotero/storage/Q329VHH3/Thorup_2007_Fully-Dynamic
Min-Cut.pdf:application/pdf},
} }