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z`o{>DkLE?jM7kz6A{HhKOGQ%>6H7@`w~_h!QQd+0@X-{BRB7T#VTFK_wZ4hJ0i3h}ZTA80b)b1^pgS4@5Iwr^KEC6CVMz>8Vq7O8%sV28 zSEMiys$deTP!uZkS2u3BPv(O!(5HtQbQm)`Q)d?^Gb7voNcP57FwQPUPA(o!X67)= R%v>DIOfcl+VhZ9g{|_s;CcXdw diff --git a/main.tex b/main.tex index 1c4d045..d13ce0b 100644 --- a/main.tex +++ b/main.tex @@ -184,6 +184,14 @@ This is the framework of the proof in \cite{arora_expander_2004}. I think the intuition behind this SDP relaxation is almost the same as \metric{}. $\ell_1$ metrics are good since they are in the cut cone. However, if we further require that the metric in \metric{} is an $\ell_1$ metric in $\R^d$, then resulting LP is NP-hard, since the integrality gap becomes 1. \cite{leighton_multicommodity_1999} showed that the $\Theta(\log n)$ gap is tight for \metric{}, but add extra constraints to \metric{} (while keeping it to be a relaxation of \scut{} and to be polynomially solvable) may provides better gap. The SDP relaxation is in fact trying to enforce the metric to be $\ell_2^2$ in $\R^n$. +\cite{arora_euclidean_2005} proved that there is an embedding from $\ell_2^2$ to $\ell_1$ with distortion $O(\sqrt{\log n}\log \log n)$. This implies an approximation for \nonuscut{} with the same ratio. $O(\sqrt{\log n})$ is likely to be the optimal bound for the above SDP. To get better gap one can stay with SDP and add more additional constraints (like Sherali-Adams, Lovász-Schrijver and Lasserre relaxations); or think distance as variables in an LP and add force feasible solution to be certain kind of metrics. \cite{arora_towards_2013} is following the former method and considers Lasserre relaxations. For the later method, getting a cut from the optimal metric is the same as embedding it to $\ell_1$. Thus it still relies on progress in metric embedding theory. Note that both methods need to satisfy +\begin{enumerate} +\item the further constrained programs is polynomially solvable, +\item it remains a relaxation of \scut{}, +\item the gap is better. +\end{enumerate} +The Lasserre relaxation of SDP automatically satisfies 1 and 2. But I believe there may be some very strange kind of metric that embeds into $\ell_1$ well? + \bibliographystyle{alpha} \bibliography{ref} \end{document} diff --git a/ref.bib b/ref.bib index 69bbb8a..aba4ac3 100644 --- a/ref.bib +++ b/ref.bib @@ -317,4 +317,20 @@ series = {SODA '95} year = {2013}, pages = {295--305}, } -@book{Williamson_Shmoys_2011, place={Cambridge}, title={The Design of Approximation Algorithms}, publisher={Cambridge University Press}, author={Williamson, David P. and Shmoys, David B.}, year={2011}} \ No newline at end of file +@book{Williamson_Shmoys_2011, place={Cambridge}, title={The Design of Approximation Algorithms}, publisher={Cambridge University Press}, author={Williamson, David P. and Shmoys, David B.}, year={2011}} + +@inproceedings{arora_towards_2013, + address = {Berkeley, CA, USA}, + title = {Towards a {Better} {Approximation} for {Sparsest} {Cut}?}, + isbn = {978-0-7695-5135-7}, + url = {http://ieeexplore.ieee.org/document/6686163/}, + doi = {10.1109/FOCS.2013.37}, + language = {en}, + urldate = {2025-05-09}, + booktitle = {2013 {IEEE} 54th {Annual} {Symposium} on {Foundations} of {Computer} {Science}}, + publisher = {IEEE}, + author = {Arora, Sanjeev and Ge, Rong and Sinop, Ali Kemal}, + month = oct, + year = {2013}, + pages = {270--279}, +}