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fix wrong math

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Yu Cong 2025-05-21 14:08:18 +08:00
parent 3ae08afcb0
commit 32ae555c29
2 changed files with 4 additions and 4 deletions
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@ -160,7 +160,7 @@ s.t.& & (x_i-x_j)^2 + (x_j-x_k)^2&\geq (x_i-x_k)^2 & &\forall i,j,k\in V\
\end{aligned} \end{aligned}
\end{equation*} \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{}. 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). Consider a relaxation which is similar to \lp{}.
\begin{equation*} \begin{equation*}
\begin{aligned} \begin{aligned}
@ -171,12 +171,12 @@ s.t.& & \sum_{ij\in V\times V}\|v_i-v_j\|^2&=1 & &\\
\end{aligned} \end{aligned}
\end{equation*} \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 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 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 there are 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 the SDP gap follows from
\[ \[
\frac{c(\delta(S))}{|S||V-S|} \frac{c(\delta(S))}{|S||V-S|}
\leq n|S| \frac{\sum_{ij\in E} c_{ij}\|v_i-v_j\|^2}{\sum_{i\in S,j\in T} \|v_i-v_j\|^2} \leq \frac{\sum_{ij\in E} c_{ij}\|v_i-v_j\|^2}{\sum_{i\in S,j\in T} \|v_i-v_j\|^2}
\leq |S| \frac{\sum_{ij\in E} c_{ij}\|v_i-v_j\|^2}{n} O(\sqrt{\log n}) \leq \frac{\sum_{ij\in E} c_{ij}\|v_i-v_j\|^2}{n^2} O(\sqrt{\log n})
\leq O(\sqrt{\log n}) \opt(SDP). \leq O(\sqrt{\log n}) \opt(SDP).
\] \]