z

.zed/tasks.json

@@ -1,25 +0,0 @@
-// Static tasks configuration.
-//
-[
-    {
-        "label": "forward_search",
-        "command": "/Applications/Skim.app/Contents/SharedSupport/displayline -r -z -b $ZED_ROW $ZED_DIRNAME/$ZED_STEM.pdf",
-        "allow_concurrent_runs": false,
-        "reveal": "never",
-        "hide": "always"
-    },
-    {
-        "label": "pdflatex_view",
-        "command": "cd \"$ZED_DIRNAME\" && pdflatex -shell-escape -synctex=-1 \"$ZED_STEM\" && /Applications/Skim.app/Contents/SharedSupport/displayline -r -z -b $ZED_ROW \"$ZED_STEM\".pdf",
-        "allow_concurrent_runs": false,
-        "reveal": "no_focus",
-        "hide": "on_success"
-    },
-    {
-        "label": "latexmk_view",
-        "command": "cd \"$ZED_DIRNAME\" && latexmk -pdf \"$ZED_STEM\" && /Applications/Skim.app/Contents/SharedSupport/displayline -r -z -b $ZED_ROW \"$ZED_STEM\".pdf",
-        "allow_concurrent_runs": false,
-        "reveal": "no_focus",
-        "hide": "on_success"
-    }
-]

main.tex

@@ -2,26 +2,19 @@
 \makeatletter
 \AfterPackage{beamerbasemodes}{\beamer@amssymbfalse}
 \makeatother
-\documentclass[aspectratio=169,handout,notheorems]{beamer}
+\documentclass[aspectratio=169,notheorems]{beamer}
 \usefonttheme{professionalfonts}
 
 \usepackage{algo}
-\usetheme{moloch}   % new metropolis
+\usetheme[block=fill]{moloch}   % new metropolis
+\setbeamercolor{block title}{bg=mDarkTeal!15}
 % fonts
-\usepackage{fontspec}
+\usepackage[defaultsans]{lato}
-\setsansfont[
-    ItalicFont={Fira Sans Italic},
-    BoldFont={Fira Sans Medium},
-    BoldItalicFont={Fira Sans Medium Italic}
-]{Fira Sans}
-\setmonofont[BoldFont={Fira Mono Medium}]{Fira Mono}
-\AtBeginEnvironment{tabular}{%
-    \addfontfeature{Numbers={Monospaced}}
-}
 \usepackage{lete-sans-math}
 \usepackage{soul}
 \usepackage[dvipsnames]{xcolor}
 \usepackage{booktabs}
+\usepackage{blkarray}
 \newtheorem{theorem}{Theorem}[section]
 \newtheorem{lemma}{Lemma}[section]
 \newtheorem{corollary}{Corollary}[section]
@@ -34,11 +27,12 @@
 \newcommand{\propositionautorefname}{Proposition}
 \newcommand{\problemautorefname}{Problem}
 \def\etal{\emph{et~al.}}
-\def\Real{\mathbb{R}}
+\def\R{\mathbb{R}}
-\def\Integer{\mathbb{Z}}
+\def\Z{\mathbb{Z}}
-\let\R\Real
+\def\F{\mathbb{F}}
-\let\Z\Integer
+\def\set#1{\left\{ #1 \right\}}
 \newcommand{\e}{\varepsilon}
+\newcommand{\p}{\pause\medskip}
 
 \title{Connectivity Interdiction \& \\ Perturbed Graphic Matroid Cogirth}
 \date{\today}
@@ -50,16 +44,16 @@
 \maketitle
 \section{Connectivity interdiction}
 \begin{frame}{Connectivity interdiction}
-\begin{problem}[connectivity interdiction {[Zenklusen ORL'14]}]
+\begin{problem}[connectivity interdiction {[Zenklusen, ORL'14]}]
 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.
 
-Find $F\subset E$ with $w(F)\leq B$ s.t. the min-cut in $G-F$ is minimized.
+Find $F\subset E$ with $c(F)\leq B$ s.t. the edge connectivity in $G-F$ is minimized.
 \end{problem}
 
-\pause\medskip
+\p
 \begin{problem}[$B$-free min-cut]
 Given the same input, define weight function for cuts
-\[ w'(\delta(X))=\min_{F\subset \delta(X),c(F)\leq B} w(\delta(X)-F)\]
+\[ w'(\delta(X))=\min_{\substack{F\subset \delta(X)\\ c(F)\leq B}} w(\delta(X)-F)\]
 
 Find the cut with minimum weight $w'$.
 \end{problem}
@@ -67,50 +61,64 @@ Find the cut with minimum weight $w'$.
 
 \begin{frame}{Applications}
 \begin{itemize}
-\item drag delivery
+\item drag delivery interdiction
-\item nuclear smuggling
+\item nuclear smuggling interdiction
 \item hospital infection control
 \item ...
 \end{itemize}
 \end{frame}
 
 \begin{frame}{Previous works}
+\newcommand{\blueT}{\textcolor{blue}{T}}
 \begin{table}[h]
 \centering
 \begin{tabular}{c c c c}
 \toprule
-unit cost $w(\cdot)=1$ & general case & random? & ref \\
+Unit cost $\textcolor{gray}{c(\cdot)=1}$ & General case & Random? & Reference \\
 \midrule
-$O(m^2n^4\log n)$ & $O(m^{2+1/\e}n^4\log n)$ & $\times$  & [Zenklusen ORL'14] \\
+$O(m^2n^4\log n)$ & $O(m^{2+1/\e}n^4\log n)$ & $\times$  & [Zenklusen, ORL'14] \\
-$O(mn^4\log^2 n)$ & $O(m^{1+1/\e}n^4\log^2 n)$ & $\checkmark$ & [Zenklusen ORL'14] \\
+$\tilde O(m+n^4 B)$ & $O(n^4\log(Bnw_{\max})\blueT)$ & $\times$ &   [Huang \etal{}, IPCO'24]\\
-$\tilde O(m+n^4 B)$ & $O(\frac{m^2n^4}{\e}\log(Bnw_{\max}))$ & $\times$ &   [Huang \etal{} IPCO'24]\\
+$\tilde O(m^2+n^3B)$ & $\tilde O(m^2+n^3\blueT)$ & $\times$ &  this work\\
-$\tilde O(n^3m\log w_{\max})$ & - & $\checkmark$ &   [Drange \etal{} AAAI'26]\\
+\midrule
-$\tilde O(m^2+n^3B)$ & $\tilde O(m^2+mn^3+\frac{n^3}{\e})$ & $\times$ &  this work\\
+$O(mn^4\log^2 n)$ & $O(m^{1+1/\e}n^4\log^2 n)$ & $\checkmark$ & [Zenklusen, ORL'14] \\
-$\tilde O(m+n^3B)$ & $\tilde O(mn^3+\frac{n^3}{\e})$ & $\checkmark$ &  this work\\
+$\tilde O(mn^3\log w_{\max})$ &   & $\checkmark$ &   [Drange \etal{}, AAAI'26]\\
+$\tilde O(m+n^3B)$ & $\tilde O(m+n^3\blueT)$ & $\checkmark$ &  this work\\
 \bottomrule
 \end{tabular}
 \caption{PTASes for connectivity interdiction}
 \end{table}
+$\blueT$ is the complexity of FPTAS for 0-1 knapsack.
 \end{frame}
 
-\begin{frame}{Method in [Huang \etal{} IPCO'24]}
+\begin{frame}{Method in [Huang \etal{}, IPCO'24]}
 
-\begin{problem}[\emph{normalized min-cut} {[Chalermsook \etal{} ICALP'22]}]
+\begin{problem}[\emph{normalized min-cut} {[Chalermsook \etal{}, ICALP'22]}]
 Given the same input as connectivity interdiction, find
 \[\argmin_{\text{cut $C$}, F\subset C} \frac{w(C-F)}{B-c(F)+1} \quad s.t. \quad 0\leq c(F)\leq B.\]
 \end{problem}
 
-\pause\medskip
+\p
-First considered in [Chalermsook \etal{} ICALP'22] as a subproblem in MWU framework when solving minimum $k$-edge connected spanning subgraph (ECSS) problem.
+... first appear in [Chalermsook \etal{}, ICALP'22] as a subproblem in MWU framework when solving some positive covering LP\footnote{minimum $k$-edge connected spanning subgraph}.
 \end{frame}
 
-\begin{frame}{Method in [Huang \etal{} IPCO'24]}
+\begin{frame}{Method in [Huang \etal{}, IPCO'24]}
-Let $\tau$ be a $(1+\e)$-approximation to the normalized min-cut and let $C^*$ be the optimal $B$-free cut.
+$\tau$ is a $(1+\e)$-approximation to the normalized min-cut\\
-\begin{lemma}
+$C^*$ is the optimal $B$-free cut.
-$C^*$ is a $(2+2\e)$-approximate min-cut in $(G,w_\tau)$, 
 
+\p
+\begin{lemma}
+There is an edge weight $w_\tau$ such that\\
+$C^*$ is a $(2+2\e)$-approximate min-cut in $(G,w_\tau)$,
 where $w_\tau(e)=\min(\tau c(e),w(e))$ is the truncated edge weight.
 \end{lemma}
+
+\p
+\begin{algo}
+enumerate approx solutions to normalized min-cut \quad  \textcolor{gray}{$\log_{1+\e}(Bnw_{\max})$}\\
+\quad   reweight the graph \quad \textcolor{gray}{$O(m)$}\\
+\quad   enumerate all $2+2\e$ min-cuts \quad \textcolor{gray}{$\tilde O(n^4)$}\\
+\quad\quad  run FPTAS for knapsack on the cut
+\end{algo}
 \end{frame}
 
 \begin{frame}{LP method}
@@ -134,7 +142,7 @@ LD=\max_{\lambda \geq 0} \min_{\text{cut $C$ and }F\subset C} w(C-F)-\lambda(B-c
 
 We are interested in $L(\lambda)=\min\limits_{\text{cut $C$ and }F\subset C} w(C)-w(F)+\lambda c(F)$.
 
-\pause \medskip
+\p
 Let $C^*$ be the optimal $B$-free mincut and let $\lambda^*$ be the optimal solution to $LD$.
 \begin{lemma}%
 \[ L(\lambda^*) \leq w_{\lambda^*}(C^*)<2L(\lambda^*)\]
@@ -143,9 +151,9 @@ Let $C^*$ be the optimal $B$-free mincut and let $\lambda^*$ be the optimal solu
 
 \section{Cogirth of perturbed graphic matroids}
 \begin{frame}{Matroid}
-A matroid $M=(E,\mathcal B)$ is a structure on set $E$.
+A \emph{matroid} $M=(E,\mathcal B)$ is a structure on set $E$.
 
-$\mathcal B$ is the collection of ``bases'' with the following properties:
+``Bases'' $\mathcal B$ is a collection of subsets with the following properties:
 \begin{itemize}
 \item $\mathcal B\neq \emptyset$;
 \item If $A$ and $B$ are distinct members of $\mathcal B$ and $a\in A-B$, 
@@ -153,22 +161,22 @@ $\mathcal B$ is the collection of ``bases'' with the following properties:
 then there exists $b\in B-A$ such that $A-a+b\in \mathcal B$.
 \end{itemize}
 
-\pause\medskip
+\p
-$X\subset E$ is a cocycle if $X\cap B\neq \emptyset$ holds for all $B\in \mathcal B$.
+$X\subset E$ is a \emph{cocycle} if $X\cap B$ is not empty for all $B\in \mathcal B$.
 
-The size of minimum cocycle is the cogirth.
+The size of minimum cocycle is the \emph{cogirth}.
 \end{frame}
 
 \begin{frame}{Examples}
 \begin{itemize}
 \item Graphic matroids. $E$ is the edge set. $\mathcal B$ is the collection of all spanning forests. Cogirth is the size of min-cut.
-\item Uniform matroids. $E$ is a set. $\mathcal B$ is the collection of all subsets of $E$ with $5$ elements. Cogirth is $|E|-4$.
+\item Uniform matroids. $E$ is a large set. $\mathcal B$ is the collection of all subsets of $E$ with $5$ elements. Cogirth is $|E|-4$.
 \item Binary matroids. $E$ is a set of binary vectors. $\mathcal B$ is the collection of maximum linearly independent sets. What is the cogirth?
 \item ...
 \end{itemize}
 \end{frame}
 
-\begin{frame}{Computing cogirth}
+\begin{frame}{Computing (co)girth}
 \[
 \small
 \begin{array}{ccccccc}
@@ -177,12 +185,90 @@ P & & P & & P & & \textcolor{BrickRed}{\text{NP-Hard}}
 \end{array}
 \]
 
-\pause\medskip
+\p
-\begin{conjecture}[{[Geelen \etal{} Ann. Comb. 2015]}]
+\begin{conjecture}[{[Geelen \etal{}, Ann. Comb. 2015]}]
-For any proper minor closed class $\mathcal M$ of binary matroids, there is a polynomial time algorithm for computing the girth of matroids in $\mathcal M$.
+For any proper minor-closed class $\mathcal M$ of binary matroids, there is a polynomial time algorithm for computing the girth of matroids in $\mathcal M$.
 \end{conjecture}
 \end{frame}
 
 \begin{frame}{Perturbed graphic matroid}
+\begin{theorem}[{[Geelen \etal{}, Ann. Comb. 2015]}]
+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 $M(A+P)$.
+\end{theorem}
+
+\p
+\begin{theorem}[{[Geelen \& Kapadia, Combinatorica'17]}]
+There are polynomial-time randomized algorithms for computing the girth and the cogirth of $M(A+P)$.
+\end{theorem}
+
+\p
+Is there a polynomial-time deterministic algorithm ?
+
+\p
+We solve the cogirth part.
 \end{frame}
+
+\begin{frame}{$(1,t)$-signed grafts}
+Geelen \& Kapadia reduce the cogirth problem of PGMs to binary matroids $M(A)$ with the following representation,
+\[
+A=
+\begin{blockarray}{ccc}
+        &   \textcolor{blue}{E(G)}    &   \textcolor{blue}{T}\\
+\begin{block}{c(cc)}
+\textcolor{blue}{V(G)}    &   A(G)    &   B\\
+\textcolor{blue}{\set{s}} &   C       &   D\\
+\end{block}
+\end{blockarray}
+\in \F_2^{(V(G)+s)\times (E(G)\cup T)}
+\]
+where $A(G)$ is the incidence matrix of a graph $G$, $T$ indexes $t$ new columns and $\set{s}$ indexes one additional row.
+
+\medskip
+$M(A)$ is a \emph{$(1,t)$-signed graft}.
+\end{frame}
+
+\begin{frame}{Previous works}
+Results on computing the cogirth of $(1,t)$-signed grafts:
+\begin{itemize}
+\item $O(r^5n)$ random algorithm [Geelen \& Kapadia, Combinatorica'17]
+\item $n^{O(1)}$ deterministic algorithm when the $\set{s}$-indexed row is all 0 [Nägele \etal{}, SODA'18]
+\end{itemize}
+\end{frame}
+
+\begin{frame}{Method}
+\begin{theorem}[{[Chekuri \etal{}, SOSA'19]}]
+Given a matroid $M$, let $\lambda(M)$ be its cogirth and let $\sigma(M)$ be the fractional base packing number.
+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.
+\end{theorem}
+
+\p
+\begin{theorem}
+If $M$ is a $(1,t)$-signed graft, then $\frac{\lambda(M)}{\sigma(M)}$ is $O(2^t)$.
+\end{theorem}
+\end{frame}
+
+\begin{frame}{Proof sketch of constant ratio}
+\begin{lemma}
+Let $M(B)$ be a binary matroid with constant ratio $\frac{\lambda(M)}{\sigma(M)}$.
+
+The ratio $\frac{\lambda(M')}{\sigma(M')}$ is also constant for $M'=M\left(\begin{bmatrix}B\\ \sigma\end{bmatrix}\right)$
+\end{lemma}
+
+\p
+\begin{lemma}
+Let $M$ be a binary matroid with representation $A\in \F_2^{n\times m}$.
+
+Let $A'$ be the binary matrix $[A,\tau]$ for any binary vector $\tau\in \F_2^n$. 
+
+If $M$ and deletion minors of $M$ have constant gap, then $M(A')/\tau$ has constant gap.
+\end{lemma}
+\end{frame}
+
+% \section{Conclusion}
+% \begin{frame}{Takeaways}
+% \begin{itemize}
+% \item try LP methods whenever possible \pause
+% \item Look at the easy cases/minors first: many theorems want to generalize
+% \end{itemize}
+% \end{frame}
 \end{document}