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Yu Cong
2025-11-27 11:31:59 +08:00
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\maketitle \maketitle
\section{Introduction} \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 design randomized polynomial time algorithms for computing girth and cogirth of perturbed graphic matroids \cite{geelen_computing_2018}. They leave an open problem for the existence of deterministic polynomial time algorithms. We solve the cogirth part using base packing techniques.
% We want to use basepacking on perturbed graphic matroid. Basepacking works for deletion closed matroid classes with constant cogirth-packing gap. % 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 $r$. 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 $r$.
@@ -35,10 +35,10 @@ A=
\end{array}, \end{array},
\] \]
where $T$ indexes $t$ new columns and $\set{s}$ indexes a new row. where $T$ indexes $t$ new columns and $\set{s}$ indexes a new row.
The matroid $M(A)$ is the linear matroid on the matrix $A\in \F_2^{(V(G)+S)\times (E(G)+T)}$. The matroid $M(A)$ is the binary matroid on the matrix $A\in \F_2^{(V(G)+S)\times (E(G)+T)}$.
\subsection{previous works} \subsection{previous works}
The cogirth problem on $(1,t)$-signed-grafts can be considered as variation of graph min-cut under congruency constraints. The cogirth problem on $(1,t)$-signed-grafts can be considered as a variation of graph min-cut under congruency constraints.
\begin{problem}[$t$-dimensional even cut, \cite{geelen_computing_2018}] \begin{problem}[$t$-dimensional even cut, \cite{geelen_computing_2018}]
\label{prob:tdimevencut} \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: 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:
@@ -53,7 +53,7 @@ This observation suggests that one cannot solve the two cases separately in poly
Another interesting special case is that $C=\emptyset$ and $D=0$. The problem becomes graph min-cut with congruency constrants, 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}. 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. They show that SFMC with constant number of constraints can be solved deterministically 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} \subsection{proof outline}
We prove that if $M(A)$ is a binary matroid on $(1,t)$-signed-graft and $t$ is a constant then $M(A)/T$ has constant cogirth-packing gap, We prove that if $M(A)$ is a binary matroid on $(1,t)$-signed-graft and $t$ is a constant then $M(A)/T$ has constant cogirth-packing gap,