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remove zed tasks. remove takeaways. fix things
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Yu Cong
2026-01-14 16:07:42 +08:00
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@@ -97,16 +97,27 @@ Given the same input as connectivity interdiction, find
\end{problem}
\p
First considered in [Chalermsook \etal{}, ICALP'22] as a subproblem in MWU framework when solving minimum $k$-edge connected spanning subgraph (ECSS) problem.
... first appear in [Chalermsook \etal{}, ICALP'22] as a subproblem in MWU framework when solving some positive covering LP\footnote{minimum $k$-edge connected spanning subgraph}.
\end{frame}
\begin{frame}{Method in [Huang \etal{}, IPCO'24]}
Let $\tau$ be a $(1+\e)$-approximation to the normalized min-cut and let $C^*$ be the optimal $B$-free cut.
\begin{lemma}
$C^*$ is a $(2+2\e)$-approximate min-cut in $(G,w_\tau)$,
$\tau$ is a $(1+\e)$-approximation to the normalized min-cut\\
$C^*$ is the optimal $B$-free cut.
\p
\begin{lemma}
There is an edge weight $w_\tau$ such that\\
$C^*$ is a $(2+2\e)$-approximate min-cut in $(G,w_\tau)$,
where $w_\tau(e)=\min(\tau c(e),w(e))$ is the truncated edge weight.
\end{lemma}
\p
\begin{algo}
enumerate approx solutions to normalized min-cut \quad \textcolor{gray}{$\log_{1+\e}(Bnw_{\max})$}\\
\quad reweight the graph \quad \textcolor{gray}{$O(m)$}\\
\quad enumerate all $2+2\e$ min-cuts \quad \textcolor{gray}{$\tilde O(n^4)$}\\
\quad\quad run FPTAS for knapsack on the cut
\end{algo}
\end{frame}
\begin{frame}{LP method}
@@ -139,7 +150,7 @@ Let $C^*$ be the optimal $B$-free mincut and let $\lambda^*$ be the optimal solu
\section{Cogirth of perturbed graphic matroids}
\begin{frame}{Matroid}
A matroid $M=(E,\mathcal B)$ is a structure on set $E$.
A \emph{matroid} $M=(E,\mathcal B)$ is a structure on set $E$.
``Bases'' $\mathcal B$ is a collection of subsets with the following properties:
\begin{itemize}
@@ -150,15 +161,15 @@ then there exists $b\in B-A$ such that $A-a+b\in \mathcal B$.
\end{itemize}
\p
$X\subset E$ is a cocycle if $X\cap B$ is not empty for all $B\in \mathcal B$.
$X\subset E$ is a \emph{cocycle} if $X\cap B$ is not empty for all $B\in \mathcal B$.
The size of minimum cocycle is the cogirth.
The size of minimum cocycle is the \emph{cogirth}.
\end{frame}
\begin{frame}{Examples}
\begin{itemize}
\item Graphic matroids. $E$ is the edge set. $\mathcal B$ is the collection of all spanning forests. Cogirth is the size of min-cut.
\item Uniform matroids. $E$ is a set. $\mathcal B$ is the collection of all subsets of $E$ with $5$ elements. Cogirth is $|E|-4$.
\item Uniform matroids. $E$ is a large set. $\mathcal B$ is the collection of all subsets of $E$ with $5$ elements. Cogirth is $|E|-4$.
\item Binary matroids. $E$ is a set of binary vectors. $\mathcal B$ is the collection of maximum linearly independent sets. What is the cogirth?
\item ...
\end{itemize}
@@ -253,10 +264,10 @@ If $M$ and deletion minors of $M$ have constant gap, then $M(A')/\tau$ has const
\end{frame}
% \section{Conclusion}
\begin{frame}{Takeaways}
\begin{itemize}
\item try LP methods whenever possible \pause
\item Look at the easy cases/minors first: many theorems want to generalize
\end{itemize}
\end{frame}
% \begin{frame}{Takeaways}
% \begin{itemize}
% \item try LP methods whenever possible \pause
% \item Look at the easy cases/minors first: many theorems want to generalize
% \end{itemize}
% \end{frame}
\end{document}