From d37c54f387d8ca02915226cf52a9414ce465a80d Mon Sep 17 00:00:00 2001 From: Yu Cong Date: Thu, 27 Nov 2025 11:26:59 +0800 Subject: [PATCH] z --- notes.tex | 44 +++++++++++++++++++++++++++++--------------- ref.bib | 2 ++ 2 files changed, 31 insertions(+), 15 deletions(-) diff --git a/notes.tex b/notes.tex index 6926e72..58ba5cf 100644 --- a/notes.tex +++ b/notes.tex @@ -49,7 +49,7 @@ Let $G=(V,E)$ be a graph and let $\ell:V\to \F_2^{t}$ be a $t$-dimensional color \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. +This observation suggests that one cannot solve the two cases separately in polynomial time. 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}. @@ -64,6 +64,19 @@ We first prove that for any binary matroid with constant gap, adding another row Contracting a set $T$ from a matroid $M$ can be done by contracting elements in $T$ one by one, $M/e_1/\dots/e_t$. We fix an arbitrary order for elements in $T$ and let $T_i$ be the set of first $i$ elements. We prove that if $(M/T_i)|E(G)$ has constant gap then so does $(M/T_{i+1})|E(G)$. \section{Proof of constant gap} + +\paragraph{notations and useful lemmas} +Let $M=(E,\mathcal I)$ be a matroid. We write $\lambda(M)$ for the cogirth of $M$ and write $\sigma(M)$ for the strength of $M$. +It is known that the strength of $M$ equals the maximum fractional base packing number and $\sigma(M)=\min_{F\subset E} \frac{|E-F|}{r(E)-r(F)}$. + +In the proof we frequently use the following result on binary matroids. +The \emph{cocircuit space} of a binary matroid $M=(E,\mathcal I)$ is the subspace of $\F_2^{E}$ that are generated by the incidence vectors of cocircuits of $M$. +\begin{lemma}[\cite{Oxley_2011}, Chapter 9] +Let $M=(E,\mathcal I)$ be a binary matroid and let $A\in \F_2^{r(M)\times |E|}$ be its binary representation. +Then the row space of $A$ equals the cocircuit space of $M$. +Any non-zero vector in the cocircuit space represents a dependent set of $M$. +\end{lemma} + We first show that adding one additional row keeps the gap constant. \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 $\sigma\in \F_2^{m}$. @@ -77,10 +90,10 @@ We can assume that $r(E)-r(F^*)\geq 1$ since otherwise we have $r'(E)-r'(F^*)\le \begin{equation*} \begin{aligned} -\sigma(M') &= \frac{|E-F^*|}{r'(E)-r'(F^*)}\\ - &\geq \frac{|E-F^*|}{r(E)+1-r(F^*)}\\ - &\geq \frac{|E-F^*|}{2(r(E)-r(F^*))}\\ - &\geq \frac{\sigma(M)}{2} +\sigma(M') &= \frac{|E-F^*|}{r'(E)-r'(F^*)} + \geq \frac{|E-F^*|}{r(E)+1-r(F^*)}\\ + &\geq \frac{|E-F^*|}{2(r(E)-r(F^*))} + \geq \frac{\sigma(M)}{2} \end{aligned} \end{equation*} Thus we have $\frac{\lambda(M')}{\sigma(M')}\leq \frac{2\lambda(M)}{\sigma(M)}$. @@ -95,14 +108,16 @@ If $M$ and deletion minors of $M$ have constant gap, then $M(A')/\tau$ has const \paragraph{Graphic case} Before proving this lemma consider an easier case where $M$ is graphic. The new element $\tau$ identifies a vertex set $T$. -A cut $\delta(X)$ is even if $|X\cap T|$ is even. The minimum cocircuit of $M(A')/\tau$ is the minimum even cut in $(G,T)$. +Note that for a cut $\delta(X)$ one can sum up the rows labelled by $v\in X$ in the incidence matrix and get an indicator vector of $\delta(X)$. +A cut $\delta(X)$ is even if $|X\cap T|$ is even. +Then the minimum cocircuit of $M(A')/\tau$ is the minimum even cut in $(G,T)$. +Let $G=(V,E)$ be a graph and let $T\subset V$ be a vertex set with even size. +An edge set $F\subset E$ is a \emph{$T$-join} if $X$ induces a subgraph in which the set of vertices with odd degree is exactly $T$. +An cut $\delta(X)$ is a \emph{$T$-cut} if $|X\cap T|$ odd. Note that any $T$-cut hits every $T$-join. For proofs, we refer to Section~12.4 in \cite{Korte_Vygen_2018}. For a spanning tree $B$, let $C(B,\tau)$ be the fundamental circuit in $B\cup \set{\tau}$. Then $C(B,\tau)\cap E$ is a $T$-join, since it induces a subgraph in which the set of vertices with odd degree is exactly $T$. - -% Now we try to characterize the set of bases of $M(A')/\tau$. -% A graph contains a $T$-join if and only if every component contains an even number of vertices in $T$. One can see that the set of bases is $\set{B-e}$ for all spanning tree $B$ and all $e\in C(B,\tau)\cap B$. -Note that the minimum base hitting set of $M(A')/t$ is the minimum even cut of $(G,T)$. +% Note that the minimum base hitting set of $M(A')/t$ is the minimum even cut of $(G,T)$. To show that gap we prove the followings, @@ -117,14 +132,14 @@ The second inequality holds since the integrality gap for graph $k$-cut is 2. For the last inequality, we claim that $x=\chi_J + \sigma(M(A')/\tau)|_{E\setminus J}$ is a feasible solution to 3-cut LP. First it is easy to see that $x(e)\leq 1$ for any edge $e$. Consider the size of $J\cap B$ for a spanning tree $B$. If this is at least 2, then $x$ satisfies the 3-cut constraint for $B$; Otherwise, $B\setminus J$ should be a base of $M'/\tau$ since $T$-cut intersects every $T$-join, and we have $\sigma(M'/T)|_{B\setminus J}\geq 1$ as a constraint in the base hitting set LP of $M(A')/\tau$. -To finish the proof consider the connections in the minimum cut of $G$, the minimum $T$-cut of $G$ and the minimum even cut of $G$. Note that if $\delta(X)$ is a $T$-cut then $|X\cap T|$ is odd. Then the minimum cut is either an $T$-cut or an even cut. +To finish the proof consider the connections in the minimum cut of $G$, the minimum $T$-cut of $G$ and the minimum even cut of $G$. Recall that the minimum $T$-cut is the minimum cut $\delta(X)$ with $|X\cap T|$ odd. Then the minimum cut is either a $T$-cut or an even cut. \begin{enumerate} \item If $J$ is the mincut. $\lambda(M'/t)\leq 2(\lambda(G)+\sigma(M'/t))\leq 2(2\sigma(G)+\sigma(M'/t)) \leq 6\sigma(M'/t)$ \item If $\lambda(M'/t)$ is the mincut. $\lambda(M'/t)=\lambda(G)\leq 2\sigma(G)\leq 2\sigma(M'/t)$ \end{enumerate} For either case, we have constant gap. -Now we generalize this idea to binary matroids. Before the proof we need some useful lemmas. +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 $M$. \begin{lemma}[$(1,d)$-good implies $(k,kd)$-good] @@ -134,7 +149,7 @@ Note that we need constant gap for all deletion minors of $M$ since we are going Instead of $T$-cut we find the minimum hitting set of $C(B,\tau)$ for all base $B$. An useful lemma is the following. We write $M+f$ for an one element extension on the binary representation of $M$. \begin{lemma}\label{TcutTjoin} Let $M$ be a binary matroid and let $M'$ be $(M+f)/f$ for a new element $f$. -The cogirth of $M$ is either the minimum hitting set of $\{C(B,f)\setminus f|\forall B\}$ or the cogirth of $M'$. +The cogirth of $M$ is either the minimum hitting set of $\{C(B,f)\setminus f|\forall B\in\mathcal B(M)\}$ or the cogirth of $M'$. \end{lemma} \begin{proof} @@ -145,7 +160,7 @@ $f$-extension can be seen as adding a new column $v$ to the binary matrix $A$. N Any cocycle has an indicator vector $yA$ and thus we can use a row vector $y$ to represent any cocycle. We say a cocycle is even if $yv=0$ and odd if $yv=1$. -First consider the cogirth of $M'$. 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 $M'$, which is the minimum set in the cocircuit space with $yv=0$. So the cogirth of $M'$ is the minimum even cocycle. +First consider the cogirth of $M'$. It follows from definitions that the minimum hitting set of $\{B-e |\forall B\in\mathcal{B}(M),\forall e\in B\cap C(B,f)\}$ is exactly the minimum cocircuit of $M'$, which is the minimum set in the cocircuit space with $yv=0$. So the cogirth of $M'$ is the minimum even cocycle. On the other hand, 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 $A\chi_F=v$. Now we apply the following lemma which is easy to prove. % 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.} @@ -164,7 +179,6 @@ We follow the same framework as the graphic case. Let $c$ be the 2-hitting set g \begin{equation}\label{eq1} \lambda(M'/\tau)\leq \lambda_2(M)\leq c\sigma_2(M)\leq c(|J|+\sigma(M'/T)|_{E\setminus J}) \end{equation} - where $J$ is the minimum hitting set of $\{C(B,f)\setminus f|\forall B\}$ and $\lambda_2(M)$ ($\sigma_2(M)$) is the integral (fractional) 2-hitting set of bases of $M$. We then apply \autoref{TcutTjoin}. diff --git a/ref.bib b/ref.bib index 3b88a6c..e90c86c 100644 --- a/ref.bib +++ b/ref.bib @@ -30,3 +30,5 @@ year = {2019}, pages = {1351--1386}, } +@book{Oxley_2011, title={Matroid Theory}, ISBN={9780198566946}, url={https://academic.oup.com/book/34846}, DOI={10.1093/acprof:oso/9780198566946.001.0001}, publisher={Oxford University Press}, author={Oxley, James}, year={2011}, month=feb } +@book{Korte_Vygen_2018, address={Berlin, Heidelberg}, series={Algorithms and Combinatorics}, title={Combinatorial Optimization: Theory and Algorithms}, volume={21}, rights={https://www.springer.com/tdm}, ISBN={9783662560389}, url={https://link.springer.com/10.1007/978-3-662-56039-6}, DOI={10.1007/978-3-662-56039-6}, publisher={Springer Berlin Heidelberg}, author={Korte, Bernhard and Vygen, Jens}, year={2018}, collection={Algorithms and Combinatorics}, language={en} } \ No newline at end of file