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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 $c(F)\leq B$ s.t. the edge connectivity in $G-F$ is minimized.
\end{problem}

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

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}
\newcommand{\blueT}{\textcolor{blue}{T}}
\begin{table}[h]
\centering
\begin{tabular}{c c c c}
\toprule
Unit cost $\textcolor{gray}{c(\cdot)=1}$ & General case & Random? & Reference \\
\midrule
$O(m^2n^4\log n)$ & $O(m^{2+1/\e}n^4\log n)$ & $\times$  & [Zenklusen, ORL'14] \\
$\tilde O(m+n^4 B)$ & $O(n^4\log(Bnw_{\max})\blueT)$ & $\times$ &   [Huang \etal{}, IPCO'24]\\
$\tilde O(m^2+n^3B)$ & $\tilde O(m^2+n^3\blueT)$ & $\times$ &  this work\\
\midrule
$O(mn^4\log^2 n)$ & $O(m^{1+1/\e}n^4\log^2 n)$ & $\checkmark$ & [Zenklusen, ORL'14] \\
$\tilde O(mn^3\log w_{\max})$ &   & $\checkmark$ &   [Drange \etal{}, AAAI'26]\\
$\tilde O(m+n^3B)$ & $\tilde O(m+n^3\blueT)$ & $\checkmark$ &  this work\\
\bottomrule
\end{tabular}
\caption{PTASes for connectivity interdiction}
\end{table}
$\blueT$ is the complexity of FPTAS for 0-1 knapsack.
\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 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]}
$\tau$ 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}
\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 \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}
\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 \emph{cocycle} if $X\cap B$ is not empty for all $B\in \mathcal B$.

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 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}
\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'17]}]
There are polynomial-time randomized algorithms for computing the girth and the cogirth of $M(A+P)$.
\end{theorem}

\p
Is there a polynomial-time deterministic algorithm ?

\p
We solve the cogirth part.
\end{frame}

\begin{frame}{$(1,t)$-signed grafts}
Geelen \& Kapadia reduce the cogirth problem of PGMs to binary matroids $M(A)$ with the following representation,
\[
A=
\begin{blockarray}{ccc}
        &   \textcolor{blue}{E(G)}    &   \textcolor{blue}{T}\\
\begin{block}{c(cc)}
\textcolor{blue}{V(G)}    &   A(G)    &   B\\
\textcolor{blue}{\set{s}} &   C       &   D\\
\end{block}
\end{blockarray}
\in \F_2^{(V(G)+s)\times (E(G)\cup T)}
\]
where $A(G)$ is the incidence matrix of a graph $G$, $T$ indexes $t$ new columns and $\set{s}$ indexes one additional row.

\medskip
$M(A)$ is a \emph{$(1,t)$-signed graft}.
\end{frame}

\begin{frame}{Previous works}
Results on computing the cogirth of $(1,t)$-signed grafts:
\begin{itemize}
\item $O(r^5n)$ random algorithm [Geelen \& Kapadia, Combinatorica'17]
\item $n^{O(1)}$ deterministic algorithm when the $\set{s}$-indexed row is all 0 [Nägele \etal{}, SODA'18]
\end{itemize}
\end{frame}

\begin{frame}{Method}
\begin{theorem}[{[Chekuri \etal{}, SOSA'19]}]
Given a matroid $M$, let $\lambda(M)$ be its cogirth and let $\sigma(M)$ be the fractional base packing number.
If there is some constant $c$ that $\frac{\lambda(M)}{\sigma(M)}<c$, then the cogirth of $M$ can be computed deterministically in polynomial number of calls to the independence oracle.
\end{theorem}

\p
\begin{theorem}
If $M$ is a $(1,t)$-signed graft, then $\frac{\lambda(M)}{\sigma(M)}$ is $O(2^t)$.
\end{theorem}
\end{frame}

\begin{frame}{Proof sketch of constant ratio}
\begin{lemma}
Let $M(B)$ be a binary matroid with constant ratio $\frac{\lambda(M)}{\sigma(M)}$.

The ratio $\frac{\lambda(M')}{\sigma(M')}$ is also constant for $M'=M\left(\begin{bmatrix}B\\ \sigma\end{bmatrix}\right)$
\end{lemma}

\p
\begin{lemma}
Let $M$ be a binary matroid with representation $A\in \F_2^{n\times m}$.

Let $A'$ be the binary matrix $[A,\tau]$ for any binary vector $\tau\in \F_2^n$. 

If $M$ and deletion minors of $M$ have constant gap, then $M(A')/\tau$ has constant gap.
\end{lemma}
\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}
\end{document}