z

main.tex

@@ -2,7 +2,7 @@
 \makeatletter
 \AfterPackage{beamerbasemodes}{\beamer@amssymbfalse}
 \makeatother
-\documentclass[aspectratio=169,handout]{beamer}
+\documentclass[aspectratio=169,handout,notheorems]{beamer}
 \usefonttheme{professionalfonts}
 
 \usepackage{algo}
@@ -19,21 +19,25 @@
     \addfontfeature{Numbers={Monospaced}}
 }
 \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\Real{\mathbb{R}}
-\def\Proj{\mathbb{P}}
-\def\Hyper{\mathbb{H}}
 \def\Integer{\mathbb{Z}}
-\def\Natural{\mathbb{N}}
-\def\Complex{\mathbb{C}}
-\def\Rational{\mathbb{Q}}
-\let\N\Natural
-\let\Q\Rational
 \let\R\Real
 \let\Z\Integer
-\def\Rd{\Real^d}
 \newcommand{\e}{\varepsilon}
 
 \title{Connectivity Interdiction \& \\ Perturbed Graphic Matroid Cogirth}
@@ -118,9 +122,67 @@ s.t.&   &   \sum_{e\in T} x_e+y_e&\geq 1    &   &\forall \text{spanning tree $T$
 &       &   y_e,x_e&\in\{0,1\}              &   &\forall e
 \end{aligned}
 \end{equation*}
+
+The integral gap is $+\infty$!
 \end{frame}
 
-\section{Computing cogirth in perturbed graphic matroids}
-\begin{frame}{Perturbed graphic matroids}
+\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)$.
+
+\pause \medskip
+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}
+
+\pause\medskip
+$X\subset E$ 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 cogirth}
+\[
+\small
+\begin{array}{ccccccc}
+\text{Graphic matroids} & \subset & \text{Regular matroids} & \subset & \text{MFMC matroids} & \subset & \text{Binary matroids} \\
+P & & P & & P & & \textcolor{BrickRed}{\text{NP-Hard}}
+\end{array}
+\]
+
+\pause\medskip
+\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}
 \end{frame}
 \end{document}