z

main.tex

@@ -2,22 +2,13 @@
 \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
 % fonts
-\usepackage{fontspec}
-\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[defaultsans]{lato}
 \usepackage{lete-sans-math}
 \usepackage{soul}
 \usepackage[dvipsnames]{xcolor}
@@ -34,11 +25,12 @@
 \newcommand{\propositionautorefname}{Proposition}
 \newcommand{\problemautorefname}{Problem}
 \def\etal{\emph{et~al.}}
-\def\Real{\mathbb{R}}
-\def\Integer{\mathbb{Z}}
-\let\R\Real
-\let\Z\Integer
+\def\R{\mathbb{R}}
+\def\Z{\mathbb{Z}}
+\def\F{\mathbb{F}}
+\def\set#1{\left\{ #1 \right\}}
 \newcommand{\e}{\varepsilon}
+\newcommand{\p}{\pause\medskip}
 
 \title{Connectivity Interdiction \& \\ Perturbed Graphic Matroid Cogirth}
 \date{\today}
@@ -56,7 +48,7 @@ Let $G=(V,E)$ be a graph with edge weights $w:E\to\Z_+$ and edge removal costs $
 Find $F\subset E$ with $w(F)\leq B$ s.t. the min-cut 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
 \[ w'(\delta(X))=\min_{F\subset \delta(X),c(F)\leq B} w(\delta(X)-F)\]
@@ -67,8 +59,8 @@ Find the cut with minimum weight $w'$.
 
 \begin{frame}{Applications}
 \begin{itemize}
-\item drag delivery
-\item nuclear smuggling
+\item drag delivery interdiction
+\item nuclear smuggling interdiction
 \item hospital infection control
 \item ...
 \end{itemize}
@@ -85,8 +77,8 @@ $O(m^2n^4\log n)$ & $O(m^{2+1/\e}n^4\log n)$ & $\times$  & [Zenklusen ORL'14] \\
 $O(mn^4\log^2 n)$ & $O(m^{1+1/\e}n^4\log^2 n)$ & $\checkmark$ & [Zenklusen ORL'14] \\
 $\tilde O(m+n^4 B)$ & $O(\frac{m^2n^4}{\e}\log(Bnw_{\max}))$ & $\times$ &   [Huang \etal{} IPCO'24]\\
 $\tilde O(n^3m\log w_{\max})$ & - & $\checkmark$ &   [Drange \etal{} AAAI'26]\\
-$\tilde O(m^2+n^3B)$ & $\tilde O(m^2+mn^3+\frac{n^3}{\e})$ & $\times$ &  this work\\
-$\tilde O(m+n^3B)$ & $\tilde O(mn^3+\frac{n^3}{\e})$ & $\checkmark$ &  this work\\
+$\tilde O(m^2+n^3B)$ & $\tilde O(m^2+mn^3+n^3/\e)$ & $\times$ &  this work\\
+$\tilde O(m+n^3B)$ & $\tilde O(mn^3+n^3/\e)$ & $\checkmark$ &  this work\\
 \bottomrule
 \end{tabular}
 \caption{PTASes for connectivity interdiction}
@@ -100,7 +92,7 @@ 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.
 \end{frame}
 
@@ -134,7 +126,7 @@ 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}%
 \[ L(\lambda^*) \leq w_{\lambda^*}(C^*)<2L(\lambda^*)\]
@@ -153,7 +145,7 @@ $\mathcal B$ 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 $B\in \mathcal B$.
 
 The size of minimum cocycle is the cogirth.
@@ -168,7 +160,7 @@ The size of minimum cocycle is the cogirth.
 \end{itemize}
 \end{frame}
 
-\begin{frame}{Computing cogirth}
+\begin{frame}{Computing (co)girth}
 \[
 \small
 \begin{array}{ccccccc}
@@ -177,12 +169,92 @@ P & & P & & P & & \textcolor{BrickRed}{\text{NP-Hard}}
 \end{array}
 \]
 
-\pause\medskip
+\p
 \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,
+\[
+A=
+\begin{array}{ccc}
+      & \begin{array}{cc} E(G) & T \end{array} \\
+    \begin{array}{r} V(G) \\ \set{s} \end{array}
+      &
+        \begin{pmatrix}
+          A(G)  &   B \\
+          C     &   D
+        \end{pmatrix}
+\end{array}
+\in \F_2^{(V(G)+s)\times (E(G)\cup T)}
+\]
+where $A(G)$ is the incidence matrix of a graph $G$, $T$ indexes $t$ new columns and $\set{s}$ indexes one additional row.
+
+\medskip
+We say $M(A)$ is a $(1,t)$-signed grafts.
+\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}