z

.gitignore

@@ -12,5 +12,6 @@
 .vscode
 __pycache__
 notes.*
+ref.log
 !notes.tex
 !.zed/*

chao.sty

@@ -65,9 +65,9 @@
 \newaliascnt{proposition}{theorem}
 
 \newtheorem{lemma}{Lemma}
-\newtheorem{corollary}{Corollary}[section]
-\newtheorem{conjecture}{Conjecture}[section]
-\newtheorem{proposition}{Proposition}[section]
+\newtheorem{corollary}{Corollary}
+\newtheorem{conjecture}{Conjecture}
+\newtheorem{proposition}{Proposition}
 \newtheorem{problem}{Problem}
 
 \newcommand{\lemmaautorefname}{Lemma}

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}

ref.bib

@@ -0,0 +1,32 @@
+
+@article{geelen_computing_2018,
+	title = {Computing {Girth} and {Cogirth} in {Perturbed} {Graphic} {Matroids}},
+	volume = {38},
+	issn = {0209-9683, 1439-6912},
+	url = {http://link.springer.com/10.1007/s00493-016-3445-3},
+	doi = {10.1007/s00493-016-3445-3},
+	language = {en},
+	number = {1},
+	urldate = {2023-03-02},
+	journal = {Combinatorica},
+	author = {Geelen, Jim and Kapadia, Rohan},
+	month = feb,
+	year = {2018},
+	pages = {167--191},
+	file = {s00493-016-3445-3 (1).pdf:/Users/congyu/Zotero/storage/EPZ6BTDE/s00493-016-3445-3 (1).pdf:application/pdf},
+}
+
+@article{nagele_submodular_2019,
+	title = {Submodular {Minimization} {Under} {Congruency} {Constraints}},
+	volume = {39},
+	issn = {1439-6912},
+	url = {https://doi.org/10.1007/s00493-019-3900-1},
+	doi = {10.1007/s00493-019-3900-1},
+	abstract = {Submodular function minimization (SFM) is a fundamental and efficiently solvable problem in combinatorial optimization with a multitude of applications in various fields. Surprisingly, there is only very little known about constraint types under which SFM remains efficiently solvable. The arguably most relevant non-trivial constraint class for which polynomial SFM algorithms are known are parity constraints, i.e., optimizing only over sets of odd (or even) cardinality. Parity constraints capture classical combinatorial optimization problems like the odd-cut problem, and they are a key tool in a recent technique to efficiently solve integer programs with a constraint matrix whose subdeterminants are bounded by two in absolute value.},
+	number = {6},
+	journal = {Combinatorica},
+	author = {Nägele, Martin and Sudakov, Benny and Zenklusen, Rico},
+	month = dec,
+	year = {2019},
+	pages = {1351--1386},
+}