generated from sxlxc/pdflatex-note
274 lines
9.4 KiB
TeX
274 lines
9.4 KiB
TeX
\RequirePackage{scrlfile}
|
|
\makeatletter
|
|
\AfterPackage{beamerbasemodes}{\beamer@amssymbfalse}
|
|
\makeatother
|
|
\documentclass[aspectratio=169,notheorems]{beamer}
|
|
\usefonttheme{professionalfonts}
|
|
|
|
\usepackage{algo}
|
|
\usetheme[block=fill]{moloch} % new metropolis
|
|
\setbeamercolor{block title}{bg=mDarkTeal!15}
|
|
% fonts
|
|
\usepackage[defaultsans]{lato}
|
|
\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]
|
|
\newtheorem{conjecture}{Conjecture}[section]
|
|
\newtheorem{proposition}{Proposition}[section]
|
|
\newtheorem{problem}{Problem}
|
|
\newcommand{\lemmaautorefname}{Lemma}
|
|
\newcommand{\corollaryautorefname}{Corollary}
|
|
\newcommand{\conjectureautorefname}{Conjecture}
|
|
\newcommand{\propositionautorefname}{Proposition}
|
|
\newcommand{\problemautorefname}{Problem}
|
|
\def\etal{\emph{et~al.}}
|
|
\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}
|
|
\author{Cong Yu}
|
|
\institute{Algorithm \& Logic Group, UESTC}
|
|
|
|
|
|
\begin{document}
|
|
\maketitle
|
|
\section{Connectivity interdiction}
|
|
\begin{frame}{Connectivity interdiction}
|
|
\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 $c(F)\leq B$ s.t. the edge connectivity in $G-F$ is minimized.
|
|
\end{problem}
|
|
|
|
\p
|
|
\begin{problem}[$B$-free min-cut]
|
|
Given the same input, define weight function for cuts
|
|
\[ w'(\delta(X))=\min_{\substack{F\subset \delta(X)\\ c(F)\leq B}} w(\delta(X)-F)\]
|
|
|
|
Find the cut with minimum weight $w'$.
|
|
\end{problem}
|
|
\end{frame}
|
|
|
|
\begin{frame}{Applications}
|
|
\begin{itemize}
|
|
\item drag delivery interdiction
|
|
\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 $\textcolor{gray}{c(\cdot)=1}$ & General case & Random? & Reference \\
|
|
\midrule
|
|
$O(m^2n^4\log n)$ & $O(m^{2+1/\e}n^4\log n)$ & $\times$ & [Zenklusen, ORL'14] \\
|
|
$\tilde O(m+n^4 B)$ & $O(n^4\log(Bnw_{\max})\blueT)$ & $\times$ & [Huang \etal{}, IPCO'24]\\
|
|
$\tilde O(m^2+n^3B)$ & $\tilde O(m^2+n^3\blueT)$ & $\times$ & this work\\
|
|
\midrule
|
|
$O(mn^4\log^2 n)$ & $O(m^{1+1/\e}n^4\log^2 n)$ & $\checkmark$ & [Zenklusen, ORL'14] \\
|
|
$\tilde O(mn^3\log w_{\max})$ & & $\checkmark$ & [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{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}
|
|
|
|
\p
|
|
... 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]}
|
|
$\tau$ is a $(1+\e)$-approximation to the normalized min-cut\\
|
|
$C^*$ is the optimal $B$-free cut.
|
|
|
|
\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}
|
|
\begin{equation*}
|
|
\begin{aligned}
|
|
\min& & \sum_{e} x_e w(e)& & & \\
|
|
s.t.& & \sum_{e\in T} x_e+y_e&\geq 1 & &\forall \text{spanning tree $T$}\quad \text{($x+y$ is a cut)}\\
|
|
& & \sum_{e} y_e c(e) &\leq B & &\text{(budget for $F$)}\\
|
|
& & y_e,x_e&\in\{0,1\} & &\forall e
|
|
\end{aligned}
|
|
\end{equation*}
|
|
|
|
The integral gap is $+\infty$!
|
|
\end{frame}
|
|
|
|
\begin{frame}{LP method}
|
|
Consider its \st{linear relaxation} Lagrangian dual.
|
|
\[
|
|
LD=\max_{\lambda \geq 0} \min_{\text{cut $C$ and }F\subset C} w(C-F)-\lambda(B-c(F))
|
|
\]
|
|
|
|
We are interested in $L(\lambda)=\min\limits_{\text{cut $C$ and }F\subset C} w(C)-w(F)+\lambda c(F)$.
|
|
|
|
\p
|
|
Let $C^*$ be the optimal $B$-free mincut and let $\lambda^*$ be the optimal solution to $LD$.
|
|
\begin{lemma}%
|
|
\[ L(\lambda^*) \leq w_{\lambda^*}(C^*)<2L(\lambda^*)\]
|
|
\end{lemma}
|
|
\end{frame}
|
|
|
|
\section{Cogirth of perturbed graphic matroids}
|
|
\begin{frame}{Matroid}
|
|
A \emph{matroid} $M=(E,\mathcal B)$ is a structure on set $E$.
|
|
|
|
``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 $a\in A-B$,
|
|
|
|
then there exists $b\in B-A$ such that $A-a+b\in \mathcal B$.
|
|
\end{itemize}
|
|
|
|
\p
|
|
$X\subset E$ is a \emph{cocycle} if $X\cap B$ is not empty for all $B\in \mathcal B$.
|
|
|
|
The size of minimum cocycle is the \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 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 (co)girth}
|
|
\[
|
|
\small
|
|
\begin{array}{ccccccc}
|
|
\text{Graphic matroids} & \subset & \text{Regular matroids} & \subset & \text{MFMC matroids} & \subset & \text{Binary matroids} \\
|
|
P & & P & & P & & \textcolor{BrickRed}{\text{NP-Hard}}
|
|
\end{array}
|
|
\]
|
|
|
|
\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 $\mathcal M$.
|
|
\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} |