From 3ae08afcb0c02eea2d9accade675be0547517066 Mon Sep 17 00:00:00 2001 From: Yu Cong Date: Tue, 20 May 2025 18:52:48 +0800 Subject: [PATCH] fix wrong math --- main.pdf | Bin 117996 -> 118445 bytes main.tex | 9 ++++++--- 2 files changed, 6 insertions(+), 3 deletions(-) diff --git a/main.pdf b/main.pdf index a41c1b178d8c287a60a337b9a5b4665433f3b69d..f2674ec0e09479d7219728fbd1b06e7ef135965c 100644 GIT binary patch delta 18480 zcmXt;Q*b6s7jE+g6Wg|J+n!8pYhpWZY}>YNXJXs7ZJqC&zs}v-)w_08-*m61t1r_b ztFs~Nh~xk)iNtcK;2azrM2YcPl)x(;ZKur^bl;cSUG1`RO!DhKXO?uGEDrhgwv;Sq ziD|hrT^TK_RF!txvFqBK{$QN|*|^Sx%aZi2FdY=GzBhaOn!JWXmazmu!3ocm+UcCH zJ8e150M>tGN@Hu{1zh83vL+=;!}g9XQq#&&3clPQRvYx(xefsfm~P@O-QPhM)2aPX1Iu7R}tsPiCVx@KILy^Y|9L@BHINuH}? zRb9-?%nBXr&C`u>T_-n9dRf70m-_0|2f^UUNhV?@Hbpc0qcq+EG862Qe2viv|6Tn~ zYFCZ9Fr+jRVHGJvgCKNbP2jxc@$t!V2VHD%oH3aJNMKZkNE|P?Pp2u^W-{bor%qeU z`QCDlA*QTMfgBkkg{DyDHssSJ@Tc9&H z$^e%YYeopKshEH&)wq51j7y&8gPu)D2xV(kcX9FZIHHK}oG{Z?KQdM7d~3Ntw2OG! 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zXl*T82(^xO{ReFWJKtyX;dkEmopauq^E>05o7?ctc_*%|(UB_2n@9p4s<>v!Ze&3V z*X)sO$bmT5Y?fTe3+NV@Wte{t1yBe@Pz)td3S|(1&A?~!8N!H7hwv{!wfV4Yba}|&Bwba@1m@sM<51uz-#CsILwmV z@cW3Ip}~+NI(~KLl5*qAN0cjBw-MSU>HVV)H!$03O5$g z{f)o_cxcyBAw6RF*oNaF&0CrX=?OO$;F(>U3~7gVu_> zzG^opm+~!@g6JqvAEhcvH?#%fU@f5_Ns?c0g;K>%ajU`#Wo~41baG{3Z3<;> fWN%_>3UhQ}a&&ldWo8O7F*h(bH3}sqMNdWw!4H?q diff --git a/main.tex b/main.tex index 4c78528..47f9dda 100644 --- a/main.tex +++ b/main.tex @@ -171,14 +171,17 @@ s.t.& & \sum_{ij\in V\times V}\|v_i-v_j\|^2&=1 & &\\ \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. +To get a $O(\sqrt{\log n})$ (randomized) approximation algorithm we need to first solve the SDP and then round the solution to get a cut $\delta(S)$ with $c(\delta(S))=|S| \opt(SDP) O(n\sqrt{\log n})$. 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 have \[ -\frac{c(\delta(S))}{|S||V-S|}\leq \frac{\sum_{ij\in E} c_{ij}\|v_i-v_j\|^2}{\sum_{i\in S,j\in T} \|v_i-v_j\|^2}\leq ? +\frac{c(\delta(S))}{|S||V-S|} +\leq n|S| \frac{\sum_{ij\in E} c_{ij}\|v_i-v_j\|^2}{\sum_{i\in S,j\in T} \|v_i-v_j\|^2} +\leq |S| \frac{\sum_{ij\in E} c_{ij}\|v_i-v_j\|^2}{n} O(\sqrt{\log n}) +\leq O(\sqrt{\log n}) \opt(SDP). \] +This is the framework of the proof in \cite{arora_expander_2004}. -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}