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zQSpMJ1|yCtjtZ}N2E1Ot*jPcMp0w=Nl1Jfob0b_D(Rvqd*VF!+u2lVMeZ08uhxa!b z{v*-fpi=BB6r#BFXy!^?xn*1_-MbSR3ieeD7<-Y#UOTB`QZU@&Gy~|aq9(mh0D*)!;Qnp2vwEA`7ia^GqV5y diff --git a/main.tex b/main.tex index a9663ab..1d06154 100644 --- a/main.tex +++ b/main.tex @@ -68,25 +68,56 @@ a 2-approximation algorithm for sparsest cut in treewidth $k$ graph with running \scut{} is easy on trees and the flow-cut gap is 1 for trees. One explaination mentioned in \citep{sparsest_cut_notes} is that shortest path distance in trees is an $\ell_1$ metric. There are works concerning planar graphs and more generally graphs with constant genus. \citep{leighton_multicommodity_1999} provided a $\Omega(\log n)$ lowerbound for flow-cut gap for \scut{}. However, it is conjectured that the gap is $O(1)$, while currently the best upperbound is still $O(\sqrt{\log n})$ \citep{rao_small_1999}. For graphs with constant genus, \citep{lee_genus_2010} gives a $O(\sqrt{\log g})$ approximation for \scut{}, where $g$ is the genus of the input graph. For flow-cut gap in planar graphs the techniques are mainly related to metric embedding theory. -\section{The Research Design} -% Requirement : Your research design may include exact details of your design and the information should be presented in coherent paragraphs: -% Example: -% Research type: e.g. qualitative study, using primary data -% Sources and other important details -% Research methods: e.g. Questionnaire surveys and interviews -% Possible difficulties/ problems or issues worth considering -% Data analysis (the specific data analysis method) -% e.g. Using SPSS to analyze the survey data and Using NVivo to analyze the interview data (details of the method and reasons for the choice) -% The significance/ implications of the study -\paragraph{Research type:} theoretical research -\paragraph{Possible difficulties:} The technical depth of open problems in \scut{} might be larger than I expected. If I have no idea after thoroughly understanding metric embedding methods and SDP relaxation, I will immediately move to other problems. -\section{Time Table} -% Data collection: e.g. During the program and first 6 months after the program (Aug. 2023- May. 2024) -% Data analysis: June 2024- Sept. 2024 -understanding existing methods: 2 weeks.\newline -solving a problem or imporving some approximation: at most 2 months. +\section{LP} + +\begin{minipage}{0.47\linewidth} +\begin{equation}\label{IP} +\begin{aligned} +\min& & \frac{\sum_e c_e x_e}{\sum_{i} D_i y_i}& & &\\ +s.t.& & \sum_{e\in p} x_e&\geq y_i & &\forall \mathcal{P}_{s_i,t_i}, \forall i\\ + & & x_e,y_i&\in \{0,1\} +\end{aligned} +\end{equation} +\end{minipage} +\begin{minipage}{0.47\linewidth} +\begin{equation}\label{LP} +\begin{aligned} +\min& & \sum_e c_e x_e& & &\\ +s.t.& & \sum_i D_iy_i&=1 & &\\ + & & \sum_{e\in p} x_e&\geq y_i & &\forall \mathcal{P}_{s_i,t_i}, \forall i\\ + & & x_e,y_i&>0 +\end{aligned} +\end{equation} +\end{minipage} +\bigskip + +\begin{minipage}{0.47\linewidth} +\begin{equation}\label{dual} +\begin{aligned} +\max& & \lambda& & &\\ +s.t.& & \sum_{p\in\mathcal{P}_{s_i,t_i}} y_p&\geq \lambda D_i & &\forall i\\ + & & \sum_i \sum_{p\in \mathcal{P}_{s_i,t_i}, p\ni e}y_p&\geq c_e & &\forall e\\ + & & y_p&\geq 0 +\end{aligned} +\end{equation} +\end{minipage} +\begin{minipage}{0.47\linewidth} +\begin{equation}\label{metric} +\begin{aligned} +\min& & \sum_{uv\in E} c_{uv}d(u,v)& & &\\ +s.t.& & \sum_i D_i d(s_i,t_i)&=1 & &\\ + & & \text{$d$ is a metric on $V$} +\end{aligned} +\end{equation} +\end{minipage} + +\newcommand{\lp}{LP\ref{LP}} +\newcommand{\ip}{IP\ref{IP}} +\begin{enumerate} +\item \ip $\geq$ \lp. Given any feasible solution to \ip, we can +\end{enumerate} \bibliographystyle{plainnat} \bibliography{ref}