generated from sxlxc/pdflatex-note
259 lines
8.8 KiB
TeX
259 lines
8.8 KiB
TeX
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\documentclass[aspectratio=169,notheorems]{beamer}
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\title{Connectivity Interdiction \& \\ Perturbed Graphic Matroid Cogirth}
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\date{\today}
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\author{Cong Yu}
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\institute{Algorithm \& Logic Group, UESTC}
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\begin{document}
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\maketitle
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\section{Connectivity interdiction}
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\begin{frame}{Connectivity interdiction}
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\begin{problem}[connectivity interdiction {[Zenklusen, ORL'14]}]
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Let $G=(V,E)$ be a graph with edge weights $w:E\to\Z_+$ and edge removal costs $c:E\to\Z_+$ and let $B\in \mathbb Z_+$ be the budget.
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Find $F\subset E$ with $w(F)\leq B$ s.t. the min-cut in $G-F$ is minimized.
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\end{problem}
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\p
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\begin{problem}[$B$-free min-cut]
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Given the same input, define weight function for cuts
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\[ w'(\delta(X))=\min_{F\subset \delta(X),c(F)\leq B} w(\delta(X)-F)\]
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Find the cut with minimum weight $w'$.
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\end{problem}
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\end{frame}
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\begin{frame}{Applications}
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\begin{itemize}
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\item drag delivery interdiction
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\item nuclear smuggling interdiction
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\item hospital infection control
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\item ...
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\end{itemize}
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\end{frame}
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\begin{frame}{Previous works}
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\begin{table}[h]
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\centering
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\begin{tabular}{c c c c}
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\toprule
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unit cost $w(\cdot)=1$ & general case & random? & ref \\
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\midrule
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$O(m^2n^4\log n)$ & $O(m^{2+1/\e}n^4\log n)$ & $\times$ & [Zenklusen, ORL'14] \\
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$O(mn^4\log^2 n)$ & $O(m^{1+1/\e}n^4\log^2 n)$ & $\checkmark$ & [Zenklusen, ORL'14] \\
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$\tilde O(m+n^4 B)$ & $O(\frac{m^2n^4}{\e}\log(Bnw_{\max}))$ & $\times$ & [Huang \etal{}, IPCO'24]\\
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$\tilde O(mn^3\log w_{\max})$ & - & $\checkmark$ & [Drange \etal{}, AAAI'26]\\
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$\tilde O(m^2+n^3B)$ & $\tilde O(m^2+mn^3+n^3/\e)$ & $\times$ & this work\\
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$\tilde O(m+n^3B)$ & $\tilde O(mn^3+n^3/\e)$ & $\checkmark$ & this work\\
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\bottomrule
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\end{tabular}
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\caption{PTASes for connectivity interdiction}
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\end{table}
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\end{frame}
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\begin{frame}{Method in [Huang \etal{}, IPCO'24]}
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\begin{problem}[\emph{normalized min-cut} {[Chalermsook \etal{}, ICALP'22]}]
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Given the same input as connectivity interdiction, find
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\[\argmin_{\text{cut $C$}, F\subset C} \frac{w(C-F)}{B-c(F)+1} \quad s.t. \quad 0\leq c(F)\leq B.\]
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\end{problem}
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\p
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First considered in [Chalermsook \etal{}, ICALP'22] as a subproblem in MWU framework when solving minimum $k$-edge connected spanning subgraph (ECSS) problem.
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\end{frame}
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\begin{frame}{Method in [Huang \etal{}, IPCO'24]}
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Let $\tau$ be a $(1+\e)$-approximation to the normalized min-cut and let $C^*$ be the optimal $B$-free cut.
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\begin{lemma}
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$C^*$ is a $(2+2\e)$-approximate min-cut in $(G,w_\tau)$,
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where $w_\tau(e)=\min(\tau c(e),w(e))$ is the truncated edge weight.
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\end{lemma}
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\end{frame}
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\begin{frame}{LP method}
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\begin{equation*}
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\begin{aligned}
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\min& & \sum_{e} x_e w(e)& & & \\
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s.t.& & \sum_{e\in T} x_e+y_e&\geq 1 & &\forall \text{spanning tree $T$}\quad \text{($x+y$ is a cut)}\\
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& & \sum_{e} y_e c(e) &\leq B & &\text{(budget for $F$)}\\
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& & y_e,x_e&\in\{0,1\} & &\forall e
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\end{aligned}
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\end{equation*}
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The integral gap is $+\infty$!
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\end{frame}
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\begin{frame}{LP method}
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Consider its \st{linear relaxation} Lagrangian dual.
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\[
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LD=\max_{\lambda \geq 0} \min_{\text{cut $C$ and }F\subset C} w(C-F)-\lambda(B-c(F))
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\]
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We are interested in $L(\lambda)=\min\limits_{\text{cut $C$ and }F\subset C} w(C)-w(F)+\lambda c(F)$.
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\p
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Let $C^*$ be the optimal $B$-free mincut and let $\lambda^*$ be the optimal solution to $LD$.
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\begin{lemma}%
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\[ L(\lambda^*) \leq w_{\lambda^*}(C^*)<2L(\lambda^*)\]
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\end{lemma}
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\end{frame}
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\section{Cogirth of perturbed graphic matroids}
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\begin{frame}{Matroid}
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A matroid $M=(E,\mathcal B)$ is a structure on set $E$.
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``Bases'' $\mathcal B$ is a collection of subsets with the following properties:
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\begin{itemize}
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\item $\mathcal B\neq \emptyset$;
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\item If $A$ and $B$ are distinct members of $\mathcal B$ and $a\in A-B$,
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then there exists $b\in B-A$ such that $A-a+b\in \mathcal B$.
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\end{itemize}
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\p
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$X\subset E$ is a cocycle if $X\cap B$ is not empty for all $B\in \mathcal B$.
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The size of minimum cocycle is the cogirth.
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\end{frame}
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\begin{frame}{Examples}
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\begin{itemize}
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\item Graphic matroids. $E$ is the edge set. $\mathcal B$ is the collection of all spanning forests. Cogirth is the size of min-cut.
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\item Uniform matroids. $E$ is a set. $\mathcal B$ is the collection of all subsets of $E$ with $5$ elements. Cogirth is $|E|-4$.
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\item Binary matroids. $E$ is a set of binary vectors. $\mathcal B$ is the collection of maximum linearly independent sets. What is the cogirth?
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\item ...
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\end{itemize}
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\end{frame}
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\begin{frame}{Computing (co)girth}
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\[
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\small
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\begin{array}{ccccccc}
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\text{Graphic matroids} & \subset & \text{Regular matroids} & \subset & \text{MFMC matroids} & \subset & \text{Binary matroids} \\
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P & & P & & P & & \textcolor{BrickRed}{\text{NP-Hard}}
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\end{array}
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\]
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\p
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\begin{conjecture}[{[Geelen \etal{}, Ann. Comb. 2015]}]
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For any proper minor-closed class $\mathcal M$ of binary matroids, there is a polynomial time algorithm for computing the girth of matroids in $\mathcal M$.
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\end{conjecture}
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\end{frame}
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\begin{frame}{Perturbed graphic matroid}
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\begin{theorem}[{[Geelen \etal{}, Ann. Comb. 2015]}]
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For any proper minor-closed class $\mathcal M$ of binary matroids, there exists two constants $k,t\in \Z_+$ such that, for each vertically $k$-connected matroid $M\in \mathcal M$, there exist matrices $A,P\in \F_2^{r\times n}$ such that $A$ is the incidence matrix of a graph, $\mathrm{rank}(P)\leq t$, and either $M$ or $M^*$ is isomorphic to $M(A+P)$.
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\end{theorem}
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\p
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\begin{theorem}[{[Geelen \& Kapadia, Combinatorica'17]}]
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There are polynomial-time randomized algorithms for computing the girth and the cogirth of $M(A+P)$.
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\end{theorem}
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\p
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Is there a polynomial-time deterministic algorithm ?
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\p
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We solve the cogirth part.
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\end{frame}
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\begin{frame}{$(1,t)$-signed grafts}
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Geelen \& Kapadia reduce the cogirth problem of PGMs to binary matroids $M(A)$ with the following representation,
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\[
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A=
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\begin{blockarray}{ccc}
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& \textcolor{blue}{E(G)} & \textcolor{blue}{T}\\
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\begin{block}{c(cc)}
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\textcolor{blue}{V(G)} & A(G) & B\\
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\textcolor{blue}{\set{s}} & C & D\\
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\end{block}
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\end{blockarray}
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\in \F_2^{(V(G)+s)\times (E(G)\cup T)}
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\]
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where $A(G)$ is the incidence matrix of a graph $G$, $T$ indexes $t$ new columns and $\set{s}$ indexes one additional row.
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\medskip
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$M(A)$ is a \emph{$(1,t)$-signed graft}.
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\end{frame}
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\begin{frame}{Previous works}
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Results on computing the cogirth of $(1,t)$-signed grafts:
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\begin{itemize}
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\item $O(r^5n)$ random algorithm [Geelen \& Kapadia, Combinatorica'17]
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\item $n^{O(1)}$ deterministic algorithm when the $\set{s}$-indexed row is all 0 [Nägele \etal{}, SODA'18]
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\end{itemize}
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\end{frame}
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\begin{frame}{Method}
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\begin{theorem}[{[Chekuri \etal{}, SOSA'19]}]
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Given a matroid $M$, let $\lambda(M)$ be its cogirth and let $\sigma(M)$ be the fractional base packing number.
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If there is some constant $c$ that $\frac{\lambda(M)}{\sigma(M)}<c$, then the cogirth of $M$ can be computed deterministically in polynomial number of calls to the independence oracle.
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\end{theorem}
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\p
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\begin{theorem}
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If $M$ is a $(1,t)$-signed graft, then $\frac{\lambda(M)}{\sigma(M)}$ is $O(2^t)$.
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\end{theorem}
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\end{frame}
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\begin{frame}{Proof sketch of constant ratio}
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\begin{lemma}
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Let $M(B)$ be a binary matroid with constant ratio $\frac{\lambda(M)}{\sigma(M)}$.
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The ratio $\frac{\lambda(M')}{\sigma(M')}$ is also constant for $M'=M\left(\begin{bmatrix}B\\ \sigma\end{bmatrix}\right)$
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\end{lemma}
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\p
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\begin{lemma}
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Let $M$ be a binary matroid with representation $A\in \F_2^{n\times m}$.
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Let $A'$ be the binary matrix $[A,\tau]$ for any binary vector $\tau\in \F_2^n$.
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If $M$ and deletion minors of $M$ have constant gap, then $M(A')/\tau$ has constant gap.
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\end{lemma}
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\end{frame}
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% \section{Conclusion}
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\begin{frame}{Takeaways}
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\begin{itemize}
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\item try LP methods whenever possible \pause
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\item Look at the easy cases/minors first: many theorems want to generalize
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\end{itemize}
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\end{frame}
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\end{document} |