From 3b2b24cd5859f686aed645888af59202863e5bc7 Mon Sep 17 00:00:00 2001 From: Yu Cong Date: Sun, 4 May 2025 11:19:55 +0800 Subject: [PATCH] FPTAS section --- beamerthemeSimple.sty | 4 ++- images/knapsack.png | Bin 0 -> 101242 bytes main.tex | 55 ++++++++++++++++++++++++++++++++++++------ ref.bib | 4 --- 4 files changed, 51 insertions(+), 12 deletions(-) create mode 100644 images/knapsack.png diff --git a/beamerthemeSimple.sty b/beamerthemeSimple.sty index 01be7ed..728ace4 100644 --- a/beamerthemeSimple.sty +++ b/beamerthemeSimple.sty @@ -4,7 +4,9 @@ \RequirePackage[sfdefault]{FiraSans} \RequirePackage{FiraMono} \renewcommand{\rmfamily}{\sffamily} -\RequirePackage[fakebold]{firamath-otf} +% \RequirePackage[fakebold]{firamath-otf} +\usepackage[mathrm=sym]{unicode-math} +\setmathfont{Fira Math} \RequirePackage{xeCJK} \setCJKmainfont{Source Han Sans SC} diff --git a/images/knapsack.png b/images/knapsack.png new file mode 100644 index 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z8L(DK;Qr`yupmU~X_}!IOK(GPWLQR%Bm@Q9iu*H!|fhdcM}BTl>Vq9nIZL0yFJn4wjkv;VQ{(S^YzI@`F{@# z94AU7K3p{hf&Bb@IRoeG#f3*JRVh-&4>BGN4XF+dn_bJBpm1`^Hb{RHDhu%S^=)YK*$M4DpqW{(i;aynJTve$jJS%`|J@jW5ZZpQcsjiL z_U+q1px5#66%cmWf!?}}gp7uBQXZ6jVoI?{Cld7c8!)8xq?k!&vcZ+}^Yi25orZHRDzi~4UG3MI0o;ArO!y?%*F zK~C<_B2>l*o4sFRsx$S;9AF(!Z)9}02#SMZiMXS9dF*GMf>~J9mP_L zrRJ+1X0goc>x-tz$;qF!{(+5I?tzJ(ci-t{>G@yH zt?vTY_W5u6z?SS`rjeJ>gSBfR&Na)xmR4>O_06aK|6UB%xh?8^R7FJdfBy##^O-SVI~NK?B7{>Vxy KNtB8ih5SFGAmDTW literal 0 HcmV?d00001 diff --git a/main.tex b/main.tex index b49d801..e20adce 100644 --- a/main.tex +++ b/main.tex @@ -1,4 +1,5 @@ \documentclass{beamer} +\usepackage{nicefrac} \title[Edge Conn Interdiction]{Faster FPTAS for Edge Connectivity Interdiction} \date{\today} @@ -42,37 +43,77 @@ Now suppose that we want to attack the network. To what extent can we decrease t \begin{problem}[edge connectivity interdiction] The input is a graph $G=(V,E)$ with edge weights $w:E\to \Z_+$ and edge removal cost $c:E\to \Z_+$ and a budget $b\in \Z_+$. The goal is to find a interdiction set $F\subset E$ with $c(F)\leq b$ that minimizes the mincut in $G-F$. \end{problem} + +How to solve this problem if\dots +\begin{itemize} + \item the optimal $F$ is given? + \item the optimal $C$ is given? +\end{itemize} \end{frame} -\begin{frame}{Examples} +\begin{frame}{Example} \begin{figure} -Examples for containing knapsack and for unweighted easy case. +\includegraphics[width=0.8\textwidth]{images/knapsack.png} \end{figure} \end{frame} \begin{frame}{Prevous Works} Zenklusen \citep{zenklusen_connectivity_2014} first studied this problem and showed the following results: \begin{itemize} -\item A PTAS\footnote{polynomial time approximation scheme} for edge connectivity interdiction; +\item A PTAS\footnote{polynomial time approximation scheme. The running time is polynomial in the input size if $\epsilon$ is fixed.} for edge connectivity interdiction; \item A $\tilde{O}(m^2 n^4)$ algorithm for the unit cost case\footnote{$\tilde{O}$ hides polylog factors}. \end{itemize} -Later \citep{vygen_fptas_2024} discovered an FPTAS\footnote{fully PTAS} with time complexity $\tilde{O}(m^2 n^4/\epsilon)$. +Later \citep{vygen_fptas_2024} discovered an FPTAS\footnote{Fully PTAS. The running time is polynomial in both the input size and $1/\epsilon$} with time complexity $\tilde{O}(m^2 n^4/\epsilon)$. \end{frame} \section{FPTAS} -\begin{frame}{placeholder} +\begin{frame}{Intermediate Problem} +\begin{problem}[Normalized Mincut] +The input is a graph $G=(V,E)$ with edge weights $w:E\to \Z_+$ and edge removal cost $c:E\to \Z_+$ and a budget $b\in \Z_+$. Find an edge set $F\subset E$ with $c(F)\leq b$ and a cut $C$ such that $\frac{w(C-F)}{b+1-c(F)}$ is minimized. +\end{problem} +Let $\tau$ be the optimum of Normalized Mincut. Consider a truncated weight $w_\tau(e)= \min \{w(e),c(e)\tau\}$. + +\begin{theorem} +The optimal cut $C^*$ for Connectivity Interdiction is a 2-approximation of global mincut with weights $w_\tau$. +\end{theorem} +\end{frame} + +\begin{frame}{Algorithm} +\begin{algo} +\underline{\textsc{FPTAS for Connectivity Interdiction}}$(G,w,c,b)$\\ +1. estimate Normalized Mincut\\ +2. enumerate all 2-approximate mincut with weight $w_\tau$\\ +3. \quad for each cut $C$ solve a knapsack to compute $F$\\ +return $(C,F)$ with smallest objective value. +\end{algo} + +1 takes $O(\log_{1+\epsilon}(poly(n)))$ time;\newline +2 takes $O(n^4)$;\newline +3 takes $O(m^2/\epsilon)$. +\newline + +complexity: $\tilde{O}(m^2n^4/\epsilon)$. \end{frame} \section{LP Perspective} -\begin{frame}{placeholder} - +\begin{frame}{LP Method} +\citep{vygen_fptas_2024} gives a strong framework but the intuition behind is vague. +\begin{equation} +\begin{aligned} +\min& & \sum_{e} x_e w(e) & & & &\\ +s.t.& & \sum_{e\in T} x_e+y_e&\geq 1 & &\forall T & &\text{($x+y$ is a cut)}\\ +& & \sum_{e} y_e c(e) &\leq b & & & &\text{(budget for $F$)}\\ +% & & x_e&\geq y_e & &\forall e\quad(F\subset C)\\ +& & y_e,x_e&\in\{0,1\} & &\forall e & & +\end{aligned} +\end{equation} \end{frame} \begin{frame}{References} diff --git a/ref.bib b/ref.bib index 07f17d1..c3f7d5a 100644 --- a/ref.bib +++ b/ref.bib @@ -10,10 +10,8 @@ booktitle = {Integer {Programming} and {Combinatorial} {Optimization}}, publisher = {Springer Nature Switzerland}, author = {Huang, Chien-Chung and Obscura Acosta, Nidia and Yingchareonthawornchai, Sorrachai}, - editor = {Vygen, Jens and Byrka, Jarosław}, year = {2024}, doi = {10.1007/978-3-031-59835-7_16}, - note = {Series Title: Lecture Notes in Computer Science}, pages = {210--223}, } @@ -27,7 +25,5 @@ journal = {Operations Research Letters}, author = {Zenklusen, Rico}, year = {2014}, - note = {Publisher: Elsevier B.V.}, - keywords = {Approximation algorithms, Interdiction problems, Multi-objective optimization, Robust optimization}, pages = {450--454}, }