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.gitea/workflows/compile.yml

@@ -17,7 +17,6 @@ jobs:
       - uses: http://localhost:3000/sxlxc/gitea-release-action@v1
         with:
           body: ''
-          prerelease: true
           name: PDF
           token: ${{ secrets.RELEASE_TOKEN }}
           tag_name: latest

.latexmkrc

@@ -1,2 +1,2 @@
-$xelatex = 'xelatex -interaction=nonstopmode -synctex=1 %O %S';
-$pdf_mode = 5;
+$pdflatex = 'xelatex -interaction=nonstopmode -synctex=1';
+$pdf_mode = 1;

main.tex

@@ -1,13 +1,260 @@
-\documentclass{beamer}
-\usetheme{metropolis}
+\RequirePackage{scrlfile}
+\makeatletter
+\AfterPackage{beamerbasemodes}{\beamer@amssymbfalse}
+\makeatother
+\documentclass[aspectratio=169,notheorems]{beamer}
+\usefonttheme{professionalfonts}
+
+\usepackage{algo}
+\usetheme{moloch}   % new metropolis
+% fonts
+\usepackage[defaultsans]{lato}
+\usepackage{lete-sans-math}
+\usepackage{soul}
+\usepackage[dvipsnames]{xcolor}
+\usepackage{booktabs}
+\newtheorem{theorem}{Theorem}[section]
+\newtheorem{lemma}{Lemma}[section]
+\newtheorem{corollary}{Corollary}[section]
+\newtheorem{conjecture}{Conjecture}[section]
+\newtheorem{proposition}{Proposition}[section]
+\newtheorem{problem}{Problem}
+\newcommand{\lemmaautorefname}{Lemma}
+\newcommand{\corollaryautorefname}{Corollary}
+\newcommand{\conjectureautorefname}{Conjecture}
+\newcommand{\propositionautorefname}{Proposition}
+\newcommand{\problemautorefname}{Problem}
+\def\etal{\emph{et~al.}}
+\def\R{\mathbb{R}}
+\def\Z{\mathbb{Z}}
+\def\F{\mathbb{F}}
+\def\set#1{\left\{ #1 \right\}}
+\newcommand{\e}{\varepsilon}
+\newcommand{\p}{\pause\medskip}
+
 \title{Connectivity Interdiction \& \\ Perturbed Graphic Matroid Cogirth}
 \date{\today}
-\author{congyu}
-\institute{A\&L Group, UESTC}
+\author{Cong Yu}
+\institute{Algorithm \& Logic Group, UESTC}
+
+
 \begin{document}
 \maketitle
-\section{First Section}
-\begin{frame}{First Frame}
-Hello, world!
+\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'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,
+\[
+A=
+\begin{array}{ccc}
+      & \begin{array}{cc} E(G) & T \end{array} \\
+    \begin{array}{r} V(G) \\ \set{s} \end{array}
+      &
+        \begin{pmatrix}
+          A(G)  &   B \\
+          C     &   D
+        \end{pmatrix}
+\end{array}
+\in \F_2^{(V(G)+s)\times (E(G)\cup T)}
+\]
+where $A(G)$ is the incidence matrix of a graph $G$, $T$ indexes $t$ new columns and $\set{s}$ indexes one additional row.
+
+\medskip
+We say $M(A)$ is a $(1,t)$-signed grafts.
+\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}