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z|Ck`SK*#-<0Ha`(K2?iPD*7e$Y{A~`hB(ttQd=}WRpF`VVQ13Ix`P5(KTKic85OK5 z>jTt6#}b#1lEv`AePd>kwCFEK|7{dF$3rM?HqVkty-`-=-Ul+ZTGd(U3i(sQzAl?;$&)Q`ycclS(memp_7Y;lc^arBO?ntGcz<9 KnW(%N^#23$75Z@i diff --git a/main.tex b/main.tex index c603d43..4c78528 100644 --- a/main.tex +++ b/main.tex @@ -152,6 +152,33 @@ I believe the later method is more general and works for \nonuscut{}, while the \subsection{SDP $O(\sqrt{\log n})$ - \uscut{}} This $O(\sqrt{\log n})$ approximation via SDP is developed in \cite{arora_expander_2004}. This is also described in \cite[section 15.4]{Williamson_Shmoys_2011}. +\begin{equation*} +\begin{aligned} +\min& & \frac{\sum_{ij\in E}c_{ij}(x_i-x_j)^2}{\sum_{ij\in V\times V}(x_i-x_j)^2}& & &\\ +s.t.& & (x_i-x_j)^2 + (x_j-x_k)^2&\geq (x_i-x_k)^2 & &\forall i,j,k\in V\\ + & & x_i&\in \{+1,-1\} & &\forall i \in V +\end{aligned} +\end{equation*} + +This SDP models \uscut{} since every assignment of $x$ corresponds to a cut and the objective is the sparsity of the cut (up to a constant factor, but we don't care since we cannot achieve a constant factor approximation anyway). Now we consider a relaxation which is similar to \lp{}. + +\begin{equation*} +\begin{aligned} +\min& & \sum_{ij\in E}c_{ij}\|v_i-v_j\|^2& & &\\ +s.t.& & \sum_{ij\in V\times V}\|v_i-v_j\|^2&=1 & &\\ + & & \|v_i-v_j\|^2 + \|v_j-v_k\|^2&\geq \|v_i-v_k\|^2 & &\forall i,j,k\in V\\ + & & v_i&\in \R^n & &\forall i \in V +\end{aligned} +\end{equation*} + +To get a $O(\sqrt{\log n})$ (randomized) approximation algorithm we need to first solve the SDP and then round the solution to get integral $x$ with $O(\sqrt{\log n}) \opt(SDP)$ upperbound on the objective. + +\[ +\frac{c(\delta(S))}{|S||V-S|}\leq \frac{\sum_{ij\in E} c_{ij}\|v_i-v_j\|^2}{\sum_{i\in S,j\in T} \|v_i-v_j\|^2}\leq ? +\] + + +If we can find two sets $S,T\subset V$ both of size $\Omega(n)$ that are well-separated, in the sense that for any $s\in S$ and $t\in T$, $\|v_s-v_t\|^2=\Omega(1/\sqrt{\log n})$, then we can move \bibliographystyle{alpha} \bibliography{ref} \end{document}