update the proof

notes.tex

@@ -8,7 +8,7 @@
 \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. It remains to show PGMs have constant gap.
+PGMs are closed under deletion.
 
 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:
@@ -48,12 +48,12 @@ Let $F^*\in \argmin_{F\subset E} \frac{|E-F|}{r'(E)-r'(F)}$.
 So the gap $\frac{\lambda(M')}{\sigma(M')}\leq \frac{2\lambda(M)}{\sigma(M)}$.
 \end{proof}
 
-\begin{lemma}
+\begin{lemma}\label{contraction_gap}
 Let $M$ be a binary matroid with representation $A\in \F_2^{n\times m}$. Let $A'$ be the binary matrix $[A,T]$ for any $T\in \F_2^n$. If $M$ and deletion minors of $M$ have constant gap, then $M(A')/T$ also has constant gap.
 \end{lemma}
 
 \paragraph{Graphic case}
-Before proving this lemma consider a easier case where $M$ is graphic. The new element $T$ identifies a vertex set $T$ (abusing notations). The minimum cocircuit of $M'/T$ is exactly the minimum even cut in $(G,T)$. Consider the fundamental circuits $C(B,T)$. $C(B,T)\cap E$ for any spanning tree $B$ is exactly the $T$-join in $B$. What is a base of $M'/T$? $B-e$ for all spanning tree $B$ and $e\in C(B,T)\setminus \set{T}$. This base hitting set of $M'/t$ is the even cut.
+Before proving this lemma consider an easier case where $M$ is graphic. The new element $T$ identifies a vertex set $T$ (abusing notations). The minimum cocircuit of $M'/T$ is exactly the minimum even cut in $(G,T)$. Consider the fundamental circuits $C(B,T)$. $C(B,T)\cap E$ for any spanning tree $B$ is exactly the $T$-join in $B$. What is a base of $M'/T$? $B-e$ for all spanning tree $B$ and $e\in C(B,T)\setminus \set{T}$. This base hitting set of $M'/t$ is the even cut.
 We want to prove the followings:
 \[
     \lambda(M'/t)\leq \lambda_2(G)\leq 2\sigma_2(G)\leq 2(|J|+\sigma(M'/T)|_{E\setminus J}),
@@ -69,9 +69,9 @@ Then we discuss the connections in the minimum cut of $G$, the minimum $T$-cut o
 Now we generalize this idea to binary matroids. Note that we need constant gap for all deletion minors of $M$ since we are going to use the gap of 2-cocycle LP of $M$.
 
 Instead of $T$-cut we find the minimum hitting set of $C(B,T)$ for all base $B$. An useful lemma is the following.
-\begin{lemma}
+\begin{lemma}\label{TcutTjoin}
 Let $M$ be a binary matroid and let $M'$ be $(M+f)/f$ for a new element $f$ ($+$ is extension on binary representation).
-The cogirth of $M$ is either the minimum hitting set of $\{C(B,f)|\forall B\}$ or the cogirth of $M'$.
+The cogirth of $M$ is either the minimum hitting set of $\{C(B,f)\setminus f|\forall B\}$ or the cogirth of $M'$.
 \end{lemma}
 
 \begin{proof}
@@ -86,15 +86,29 @@ Let $A$ be the binary representation of $M$. Note that the cocircuit space of $M
 
 Now we consider the hitting sets. It follows from definitions that the minimum hitting set of $\{B-e |\forall B ,\forall e\in B\cap C(B,f)\}$ is exactly the minimum cocircuit of $(M+f)/f$, which is the minimum set in the cocircuit space with $yv=0$. So the minimum hitting set of $B-e$ is always an even cocycle.
 
-Consider a set $X$ which fails to hit all $\{C(B,f)|\forall B\}$. Then there must be a set $F\subseteq E\setminus X$ that span the new element $f$. On matrix, this is equivalent to the existence of $\chi_F$ that $A\chi_F=v$. 
+Consider a set $X$ which fails to hit all $\{C(B,f)\setminus f|\forall B\}$. Then there must be a set $F\subseteq E\setminus X$ that span the new element $f$. On matrix, this is equivalent to the existence of a indicator vector $\chi_F$ such that $A\chi_F=v$. 
 % Using Farkas' lemma on $\mathbb F_2$, this implies that there dose not exists $y$ that $\supp(yA)\subset X\land yv=1$. \note{chatGPT says one can use Farkas lemma on finite field like this, need to verify.}
 \begin{lemma}[Farkas' lemma on $\F_2$]
 Let $A$ be a $n\times m$ binary matrix and $b$ be a $n$-dimensional binary vector. Either there is a $x\in \F_2^m$ such that $Ax=b$, or there is $y\in \F_2^n$ such that $y^TA=0$ and $y^Tb=1$.
 \end{lemma}
 Then $X$ fails to be a hitting set is equivalent to the fact that there does not exist $y$ satisfying $\supp(yA)\subset X\land yv=1$.
-So $X$ hits all $\{C(B,f)|\forall B\}$ iff there is such a $y$. Minimizing $X$ pushes it to $\supp(yA)$ which is an odd cocycle. Hence, the minimum hitting set for $C(B,f)$ is always an odd cocycle.
+So $X$ hits all $\{C(B,f)\setminus f|\forall B\}$ iff there is such a $y$. Minimizing $X$ pushes it to $\supp(yA)$ which is an odd cocycle. Hence, the minimum hitting set for $\{C(B,f)\setminus f|\forall B\}$ is always an odd cocycle.
 
 The theorem then follows directly from the fact that any cocycle is either odd or even.
 \end{proof}
 
+Now we prove \autoref{contraction_gap}.
+\begin{proof}
+We follow the same framework as graphic case. Let $c$ be the 2-hitting set gap and let $c'$ be the cogirth-packing gap of $M$. We have
+\begin{equation}\label{eq1}
+\lambda(M'/t)\leq \lambda_2(G)\leq c\sigma_2(G)\leq c(|J|+\sigma(M'/T)|_{E\setminus J})
+\end{equation}
+
+where $J$ is the minimum hitting set of $\{C(B,f)\setminus f|\forall B\}$.
+
+We then apply \autoref{TcutTjoin}. If $\lambda(M'/t)=\lambda(M)$, then we have $\lambda(M'/t)= \lambda(M)\leq c \sigma(M)\leq c\sigma(M'/t)$, since the optimal solution to $\sigma(M'/t)$ is feasible to $\sigma(M)$. Otherwise, we extend \autoref{eq1} and get $\lambda(M'/t)\leq c(|J|+\sigma(M'/T))=c(\lambda(M)+\sigma(M'/T))\leq c(c'+1)\sigma(M'/T)$.
+\end{proof}
+
+The constant gap for matroids on $(1,t)$-signed grafts then follows.
+
 \end{document}