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+++ b/main.tex @@ -45,20 +45,15 @@ From a more mathematical perspective, the techniques developed for approximating Besides theoretical interests, \scut{} is useful in practical scenarios such as in image segmentation and in some machine leaning algorithms. \subsection{related works} -\nonuscut{} is APX-hard \cite{juliaJACMapxhard} and, assuming the Unique Game Conjecture, has no polynomial time constant factor aproximation algorithm\cite{chawla_hardness_2005}. \scut{} admits no PTAS \cite{uniformhardnessFocs07}, assuming a widely believed conjecture. The currently best approximation algorithm for \scut{} has ratio $O(\sqrt{\log n})$ and running time $\tilde{O}(n^2)$ \cite{arora_osqrtlogn_2010}. Prior to this currently optimal result, there is a long line of research optimizing both the approximation ratio and the complexity, see \cite{arora_expander_2004,leighton_multicommodity_1999}. +\nonuscut{} is APX-hard \cite{juliaJACMapxhard} and, assuming the Unique Game Conjecture, has no polynomial time constant factor aproximation algorithm\cite{chawla_hardness_2005}. \scut{} admits no PTAS \cite{uniformhardnessFocs07}, assuming that NP-complete problems cannot be solved in randomized subexponential time. The currently best approximation algorithm for \scut{} has ratio $O(\sqrt{\log n})$ and running time $\tilde{O}(n^2)$ \cite{arora_osqrtlogn_2010}. Prior to this currently optimal result, there is a long line of research optimizing both the approximation ratio and the complexity, see \cite{arora_expander_2004,leighton_multicommodity_1999}. There are also works concerning approximating \scut{} on special graph classes such as planar graphs \cite{lee_genus_2010}, graphs with low treewidth \cite{chlamtac_approximating_2010,gupta2013sparsestcutboundedtreewidth, Chalermsook_2024}. -For an overview of the LP methods for \scut{}, see \url{https://courses.grainger.illinois.edu/cs598csc/fa2024/Notes/lec-sparsest-cut.pdf}. +The seminal work of \cite{leighton_multicommodity_1999} starts this line of research. They studied multicommodity flow problem and proved a $O(\log n)$ flow-cut gap (in fact the tight $\Theta(\log n)$ gap was proven by \cite{aumann_rabani_1995} and \cite{Linial_London_Rabinovich_1995}). They also developed $O(\log n)$ approximation algorithm for multicommodity flow problems, which implies $O(\log n)$ approximation for \scut{}. The technique is called region growing. They also discovered a lowerbound of $\Omega(\log n)$ via expanders. Note that any algorithm achieving the $O(\log n)$ flow cut gap implies an $O(\log^2 n)$ approximation for \nonuscut{}, but better ratio is still possible through other methods. This paper showed that $O(\log^2 n)$ is the best approximation we can achieve using flow-cut gap. -% \subsection{open problems} - -% One major open problem for \scut{} is the best approximation ratio for planar graphs. It is conjectured that the ratio for planar graphs is $O(1)$ but currently the best lowerbound is $O(\sqrt{\log n})$. For graphs treewidth $k$, an open problem is that whether there is a 2 approximation algorithm that runs in $2^{O(k)}\poly(n)$. - -The seminal work of \cite{leighton_multicommodity_1999} starts this line of research. They studied multicommodity flow problem and proved a $O(\log n)$ flow-cut gap (in fact the tight $\Theta(\log n)$ gap was proven by \cite{aumann_rabani_1995} and \cite{Linial_London_Rabinovich_1995}). They also developed $O(\log n)$ approximation algorithm for multicommodity flow problems, which can imply $O(\log n)$ approximation for \scut{} and $O(\log^2 n)$ approximation for \nonuscut{}. The technique is called region growing. They also discovered a lowerbound of $\Omega(\log n)$ via expanders. Note that any algorithm achieving the $O(\log n)$ flow cut gap implies an $O(\log^2 n)$ approximation for \nonuscut{}, but better ratio is still possible through other methods. This paper showed that $O(\log^2 n)$ is the best approximation we can achieve using flow-cut gap. - -For \nonuscut{} \cite{leighton_multicommodity_1999} only guarantees a $O(\log^2 n)$ approximation. This is further improved by \cite{Linial_London_Rabinovich_1995} and \cite{lognGapAumann98}. \cite{lognGapAumann98} applied metric embedding to \nonuscut{} and obtained a $O(\log n)$ approximation. The connections between metric embedding and \nonuscut{} is influential. \nonuscut{} can be formulated as an integer program. \cite{lognGapAumann98}, \cite{aumann_rabani_1995} and \cite{Linial_London_Rabinovich_1995} considered the metric relaxation of the IP. They observed that \nonuscut{} is polynomial time solvable for trees and more generally for all $\ell_1$ metrics. The $O(\log n)$ approximation follows from the $O(\log n)$ distortion in the metric embedding theorem. +For \nonuscut{} \cite{leighton_multicommodity_1999} only guarantees an approximation ratio of $O(\log^2 n)$. This is further improved by \cite{Linial_London_Rabinovich_1995} and \cite{lognGapAumann98}. \cite{lognGapAumann98} applied metric embedding to \nonuscut{} and obtained a $O(\log n)$ approximation. The connections between metric embedding and \nonuscut{} is influential. \nonuscut{} can be formulated as an integer program. \cite{lognGapAumann98}, \cite{aumann_rabani_1995} and \cite{Linial_London_Rabinovich_1995} considered the metric relaxation of the IP. They observed that \nonuscut{} is polynomial time solvable for trees and more generally for all $\ell_1$ metrics. The $O(\log n)$ approximation follows from the $O(\log n)$ distortion in the metric embedding theorem. \cite{arora_expander_2004} and \cite{arora_osqrtlogn_2010} further improved the approximation ratio for \scut{} to $O(\sqrt{\log n})$ via semidefinite relaxation. This is currently the best approximation ratio for \scut{}. +For \nonuscut{}, the approximation is improved to $O(\sqrt{\log n} \log \log n)$ \cite{arora_euclidean_2005,arora_frechet_2007}. Later \cite{guruswami_approximating_2013} gives a $\frac{1+\delta}{\e}$ approximation in time $2^{r/(\delta \e)}\poly(n)$ provided that $\lambda_r\geq \opt / (1-\delta)$. There is also plenty of research concerning \scut{} on some graph classes, for example \cite{bonsma_complexity_2012}. One of the most popular class is graphs with constant treewidth. \cite{Chalermsook_2024} gave a $O(k^2)$ approximation algorithm with complexity $2^{O(k)}\poly(n)$. \cite{Cohen-Addad_Mömke_Verdugo_2024} obtained a 2-approximation algorithm for sparsest cut in treewidth $k$ graph with running time $2^{2^{O(k)}}\poly(n)$. diff --git a/ref.bib b/ref.bib index 71e2685..c48dcb2 100644 --- a/ref.bib +++ b/ref.bib @@ -243,3 +243,50 @@ numpages = {10}, location = {San Francisco, California, USA}, series = {SODA '95} } + +@article{arora_frechet_2007, + title = {Fréchet {Embeddings} of {Negative} {Type} {Metrics}}, + volume = {38}, + issn = {1432-0444}, + url = {https://doi.org/10.1007/s00454-007-9007-0}, + doi = {10.1007/s00454-007-9007-0}, + language = {en}, + number = {4}, + urldate = {2025-05-19}, + journal = {Discrete \& Computational Geometry}, + author = {Arora, Sanjeev and Lee, James R. and Naor, Assaf}, + month = dec, + year = {2007}, + keywords = {Convex and Discrete Geometry, Differential Geometry, Distortion, Euclidean, Functional Analysis, Geometry, Global Analysis and Analysis on Manifolds, Hyperbolic Geometry, L 1, Metric embeddings, Sparsest cut problem}, + pages = {726--739}, +} + +@inproceedings{arora_euclidean_2005, + address = {New York, NY, USA}, + series = {{STOC} '05}, + title = {Euclidean distortion and the sparsest cut}, + isbn = {978-1-58113-960-0}, + url = {https://doi.org/10.1145/1060590.1060673}, + doi = {10.1145/1060590.1060673}, + urldate = {2025-05-19}, + booktitle = {Proceedings of the thirty-seventh annual {ACM} symposium on {Theory} of computing}, + publisher = {Association for Computing Machinery}, + author = {Arora, Sanjeev and Lee, James R. and Naor, Assaf}, + month = may, + year = {2005}, + pages = {553--562}, +} + +@inproceedings{guruswami_approximating_2013, + address = {USA}, + series = {{SODA} '13}, + title = {Approximating non-uniform sparsest cut via generalized spectra}, + isbn = {978-1-61197-251-1}, + urldate = {2025-05-18}, + booktitle = {Proceedings of the twenty-fourth annual {ACM}-{SIAM} symposium on {Discrete} algorithms}, + publisher = {Society for Industrial and Applied Mathematics}, + author = {Guruswami, Venkatesan and Sinop, Ali Kemal}, + month = jan, + year = {2013}, + pages = {295--305}, +} \ No newline at end of file