z

notes.tex

@@ -1,34 +1,63 @@
-\documentclass[a4paper,11pt]{article}
+\documentclass[a4paper,12pt]{article}
 \usepackage{chao}
+
 \DeclareMathOperator{\supp}{supp}
+
 \title{Cogirth of Perturbed Graphic Matroids}
 \author{}
-\date{}
 \begin{document}
 \maketitle
 
-We want to use basepacking on perturbed graphic matroid. Basepacking works for deletion closed matroid classes with constant cogirth-packing gap. 
-PGMs are closed under deletion.
+\section{Introduction}
+Geelen and Kapadia design randomized polynomial time algorithms for computing girth and cogirth of perturbed graphic matroids \cite{geelen_computing_2018}. They leave an open problem that if there are deterministic polynomial time algorithms. We solve the cogirth part using base packing techniques.
 
-Geelen and Kapadia reduce the cogirth problem on PGMs to finding the cogirth of $(1,t)$-signed grafts.
-We work on binary matroid $M$ defined on the following binary matrix:
+% We want to use basepacking on perturbed graphic matroid. Basepacking works for deletion closed matroid classes with constant cogirth-packing gap. 
+A binary matroid is a low rank perturbed graphic matroid (PGM) if it has a binary representation $A+P$, where $A$ is the incidence matrix of a graph and $P$ is a binary matrix with rank at most a constant $r$.
+For fixed $r$, low rank perturbed graphic matroids are closed under minors. PGMs play a central role in the following conjecture.\footnote{Details can be found in the introduction of \cite{geelen_computing_2018}.}
+\begin{conjecture}
+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}
 
+Geelen and Kapadia reduce the cogirth problem on low rank PGMs to the one on $(1,t)$-signed-grafts.
+% We prove that matroids on $(1,t)$-signed-grafts have constant cogirth-packing gap.
+Let $G$ be a graph and let $A(G)$ be its incidence matrix.
+The incidence matrix of a $(1,t)$-signed-graft $(G,\set{s},T,B,C,D)$ is the binary matrix
 \[
+A=
 \begin{array}{ccc}
-      & \begin{array}{cc} E(G) & \tau \end{array} \\
-    \begin{array}{r} V(G) \\ \omega \end{array}
+      & \begin{array}{cc} E(G) & T \end{array} \\
+    \begin{array}{r} V(G) \\ \set{s} \end{array}
       &
         \begin{pmatrix}
-          A(G) & T \\
-          \sigma & \alpha
+          A(G)  &   B \\
+          C     &   D
         \end{pmatrix}
-\end{array}
+\end{array},
 \]
-where $G$ is a connect graph and $A(G)$ is the incidence matrix of $G$.
-The goal is to find the cogirth of $M/\tau$.
+where $T$ indexes $t$ new columns and $\set{s}$ indexes a new row.
+The matroid $M(A)$ is the linear matroid on the matrix $A\in \F_2^{(V(G)+S)\times (E(G)+T)}$.
 
-This matroid can be built from a graphic matroid via a series of extension, contraction and co-extension. We first show that $M([A,T])/T$ has constant gap, and then prove a constant gap of $M/\tau$.
+\subsection{previous works}
+The cogirth problem on $(1,t)$-signed-grafts can be considered as variation of graph min-cut under congruency constraints.
+\begin{problem}[$t$-dimensional even cut, \cite{geelen_computing_2018}]
+\label{prob:tdimevencut}
+Let $G=(V,E)$ be a graph and let $\ell:V\to \F_2^{t}$ be a $t$-dimensional coloring on vertices. Given a edge set $C\subset E$ and a coloring $D\in  \F_2^{1\times t}$, find a non-empty vertex set $X\subset V$ that minimizes the smaller value of the following two:
+\begin{enumerate}
+\item the minimum $|\delta(X)|$ such that $\sum_{v\in X}\ell(v)=0$;
+\item the minimum $|\delta(X)\setsymdiff C|$ such that $\sum_{v\in X}\ell(v)=D$.
+\end{enumerate}
+\end{problem}
 
+Consider a special case of \autoref{prob:tdimevencut} that $\ell=D=0$ for all vectices and $C=E$. Then the $\delta(X)$ achieving the minimum value of case 2 is exactly the max-cut of $G$.
+This observation suggests that one cannot deal with the two cases separately.
+
+Another interesting special case is that $C=\emptyset$ and $D=0$. The problem becomes graph min-cut with congruency constrants, 
+which is a special case of submodular function minimization under congruency constraints (SFMC) studied by Nägele \etal \cite{nagele_submodular_2019}.
+They show that SFMC with constant number of constraints can be solved in polynomial time if the modular is prime. However, the objective in case 2 of \autoref{prob:tdimevencut} is not submodular so their method does not generalize.
+
+\subsection{proof outline}
+
+\section{Proof of constant gap}
 \begin{lemma}
 Let $M$ be a binary matroid with binary representation $B\in \F_2^{n\times m}$. If $M$ has constant gap, then $M\left(\begin{bmatrix}B\\ \sigma\end{bmatrix}\right)$ has constant gap for any row vector $\sigma\in \F_2^{m}$.
 \end{lemma}
@@ -110,5 +139,6 @@ We then apply \autoref{TcutTjoin}. If $\lambda(M'/t)=\lambda(M)$, then we have $
 \end{proof}
 
 The constant gap for matroids on $(1,t)$-signed grafts then follows.
-
+\bibliographystyle{plain}
+\bibliography{ref}
 \end{document}