From 166d188806e9af6ccb2007569ccace333b454a2b Mon Sep 17 00:00:00 2001
From: Yu Cong <sxlxcsxlxc@gmail.com>
Date: Wed, 31 Dec 2025 14:35:28 +0800
Subject: [PATCH] z

---
 main.tex | 76 +++++++++++++++++++++++++++++++++++++++++++++++++++++---
 1 file changed, 73 insertions(+), 3 deletions(-)

diff --git a/main.tex b/main.tex
index bedc82b..fe27d00 100644
--- a/main.tex
+++ b/main.tex
@@ -2,7 +2,7 @@
 \makeatletter
 \AfterPackage{beamerbasemodes}{\beamer@amssymbfalse}
 \makeatother
-\documentclass[aspectratio=169,handout,notheorems]{beamer}
+\documentclass[aspectratio=169,notheorems]{beamer}
 \usefonttheme{professionalfonts}
 
 \usepackage{algo}
@@ -28,6 +28,7 @@
 \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}
 
@@ -180,11 +181,80 @@ For any proper minor-closed class $\mathcal M$ of binary matroids, there exists
 \end{theorem}
 
 \p
-\begin{theorem}[{[Geelen \& Kapadia Combinatorica'18]}]
-There are polynomial-time randomized algorithm for computing the girth and the cogirth of $M(A+P)$.
+\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}
\ No newline at end of file