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\title{Connectivity Interdiction \& \\ Perturbed Graphic Matroid Cogirth}
\date{\today}
\author{Cong Yu}
\institute{Algorithm \& Logic Group, UESTC}


\begin{document}
\maketitle
\section{Connectivity interdiction}
\begin{frame}{Connectivity interdiction}
\begin{problem}[connectivity interdiction {[Zenklusen ORL'14]}]
Let $G=(V,E)$ be a graph with edge weights $w:E\to\Z_+$ and edge removal costs $c:E\to\Z_+$ and let $B\in \mathbb Z_+$ be the budget.

Find $F\subset E$ with $w(F)\leq B$ s.t. the min-cut in $G-F$ is minimized.
\end{problem}

\p
\begin{problem}[$B$-free min-cut]
Given the same input, define weight function for cuts
\[ w'(\delta(X))=\min_{F\subset \delta(X),c(F)\leq B} w(\delta(X)-F)\]

Find the cut with minimum weight $w'$.
\end{problem}
\end{frame}

\begin{frame}{Applications}
\begin{itemize}
\item drag delivery interdiction
\item nuclear smuggling interdiction
\item hospital infection control
\item ...
\end{itemize}
\end{frame}

\begin{frame}{Previous works}
\begin{table}[h]
\centering
\begin{tabular}{c c c c}
\toprule
unit cost $w(\cdot)=1$ & general case & random? & ref \\
\midrule
$O(m^2n^4\log n)$ & $O(m^{2+1/\e}n^4\log n)$ & $\times$  & [Zenklusen ORL'14] \\
$O(mn^4\log^2 n)$ & $O(m^{1+1/\e}n^4\log^2 n)$ & $\checkmark$ & [Zenklusen ORL'14] \\
$\tilde O(m+n^4 B)$ & $O(\frac{m^2n^4}{\e}\log(Bnw_{\max}))$ & $\times$ &   [Huang \etal{} IPCO'24]\\
$\tilde O(n^3m\log w_{\max})$ & - & $\checkmark$ &   [Drange \etal{} AAAI'26]\\
$\tilde O(m^2+n^3B)$ & $\tilde O(m^2+mn^3+n^3/\e)$ & $\times$ &  this work\\
$\tilde O(m+n^3B)$ & $\tilde O(mn^3+n^3/\e)$ & $\checkmark$ &  this work\\
\bottomrule
\end{tabular}
\caption{PTASes for connectivity interdiction}
\end{table}
\end{frame}

\begin{frame}{Method in [Huang \etal{} IPCO'24]}

\begin{problem}[\emph{normalized min-cut} {[Chalermsook \etal{} ICALP'22]}]
Given the same input as connectivity interdiction, find
\[\argmin_{\text{cut $C$}, F\subset C} \frac{w(C-F)}{B-c(F)+1} \quad s.t. \quad 0\leq c(F)\leq B.\]
\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.
\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 $(G,w_\tau)$, 

where $w_\tau(e)=\min(\tau c(e),w(e))$ is the truncated edge weight.
\end{lemma}
\end{frame}

\begin{frame}{LP method}
\begin{equation*}
\begin{aligned}
\min&   &   \sum_{e} x_e w(e)&              &   & \\
s.t.&   &   \sum_{e\in T} x_e+y_e&\geq 1    &   &\forall \text{spanning tree $T$}\quad \text{($x+y$ is a cut)}\\
&       &   \sum_{e} y_e c(e) &\leq B       &   &\text{(budget for $F$)}\\
&       &   y_e,x_e&\in\{0,1\}              &   &\forall e
\end{aligned}
\end{equation*}

The integral gap is $+\infty$!
\end{frame}

\begin{frame}{LP method}
Consider its \st{linear relaxation} Lagrangian dual.
\[
LD=\max_{\lambda \geq 0} \min_{\text{cut $C$ and }F\subset C} w(C-F)-\lambda(B-c(F))
\]

We are interested in $L(\lambda)=\min\limits_{\text{cut $C$ and }F\subset C} w(C)-w(F)+\lambda c(F)$.

\p
Let $C^*$ be the optimal $B$-free mincut and let $\lambda^*$ be the optimal solution to $LD$.
\begin{lemma}%
\[ L(\lambda^*) \leq w_{\lambda^*}(C^*)<2L(\lambda^*)\]
\end{lemma}
\end{frame}

\section{Cogirth of perturbed graphic matroids}
\begin{frame}{Matroid}
A matroid $M=(E,\mathcal B)$ is a structure on set $E$.

$\mathcal B$ is the collection of ``bases'' with the following properties:
\begin{itemize}
\item $\mathcal B\neq \emptyset$;
\item If $A$ and $B$ are distinct members of $\mathcal B$ and $a\in A-B$, 

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 $X\cap B\neq \emptyset$ holds for all $B\in \mathcal B$.

The size of minimum cocycle is the 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 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}
\end{frame}

\begin{frame}{Computing (co)girth}
\[
\small
\begin{array}{ccccccc}
\text{Graphic matroids} & \subset & \text{Regular matroids} & \subset & \text{MFMC matroids} & \subset & \text{Binary matroids} \\
P & & P & & P & & \textcolor{BrickRed}{\text{NP-Hard}}
\end{array}
\]

\p
\begin{conjecture}[{[Geelen \etal{} Ann. Comb. 2015]}]
For any proper minor-closed class $\mathcal M$ of binary matroids, there is a polynomial time algorithm for computing the girth of matroids in $\mathcal M$.
\end{conjecture}
\end{frame}

\begin{frame}{Perturbed graphic matroid}
\begin{theorem}[{[Geelen \etal{} Ann. Comb. 2015]}]
For any proper minor-closed class $\mathcal M$ of binary matroids, there exists two constants $k,t\in \Z_+$ such that, for each vertically $k$-connected matroid $M\in \mathcal M$, there exist matrices $A,P\in \F_2^{r\times n}$ such that $A$ is the incidence matrix of a graph, $\mathrm{rank}(P)\leq t$, and either $M$ or $M^*$ is isomorphic to $M(A+P)$.
\end{theorem}

\p
\begin{theorem}[{[Geelen \& Kapadia Combinatorica'18]}]
There are polynomial-time randomized algorithm for computing the girth and the cogirth of $M(A+P)$.
\end{theorem}

\p
Is there a polynomial-time deterministic algorithm ?
\end{frame}
\end{document}