From 2ba95ac69b4ba367830541251552e0dff4cd7149 Mon Sep 17 00:00:00 2001 From: Yu Cong Date: Thu, 27 Nov 2025 11:31:59 +0800 Subject: [PATCH] z --- notes.tex | 8 ++++---- 1 file changed, 4 insertions(+), 4 deletions(-) diff --git a/notes.tex b/notes.tex index 58ba5cf..e8a5fd9 100644 --- a/notes.tex +++ b/notes.tex @@ -9,7 +9,7 @@ \maketitle \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. 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}, \] 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} -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}] \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: @@ -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, 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} 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,