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@@ -45,7 +45,7 @@ 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}
-\scut{} is APX-hard \citep{juliaJACMapxhard} and, assuming the Unique Game Conjecture, has no polynomial time constant factor aproximation algorithm\citep{chawla_hardness_2005}. The currently best approximation algorithm has ratio $O(\sqrt{\log n})$ and running time $\tilde{O}(n^2)$ \citep{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 \citep{arora_expander_2004,leighton_multicommodity_1999}.
+\nonuscut{} is APX-hard \citep{juliaJACMapxhard} and, assuming the Unique Game Conjecture, has no polynomial time constant factor aproximation algorithm\citep{chawla_hardness_2005}. \scut{} admits no PTAS \citep{uniformhardnessFocs07}, assuming a widely believed conjecture. The currently best approximation algorithm has ratio $O(\sqrt{\log n})$ and running time $\tilde{O}(n^2)$ \citep{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 \citep{arora_expander_2004,leighton_multicommodity_1999}.
 There are also works concerning approximating \scut{} on special graph classes such as planar graphs \citep{lee_genus_2010}, graphs with low treewidth \citep{chlamtac_approximating_2010,gupta2013sparsestcutboundedtreewidth, Chalermsook_2024}.
 
 For an overview of the LP methods for \scut{}, see \citep{sparsest_cut_notes}.
@@ -61,6 +61,13 @@ The seminal work of \cite{leighton_multicommodity_1999} starts this line of rese
 For \nonuscut{} \citep{leighton_multicommodity_1999} only guarantees a $O(\log^2 n)$ approximation. This is further improved by \citep{Linial_London_Rabinovich_1995} and \citep{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. \citeauthor{lognGapAumann98} 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.
 
 \citep{arora_expander_2004} and \citep{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{}.
+
+There is also plenty of research concerning \scut{} on some graph classes, for example \citep{bonsma_complexity_2012}. One of the most popular class is graphs with constant treewidth. \citep{Chalermsook_2024} gave a $O(k^2)$ approximation algorithm with complexity $2^{O(k)}\poly(n)$. \citep{Cohen-Addad_Mömke_Verdugo_2024} obtained
+a 2-approximation algorithm for sparsest cut in treewidth $k$ graph with running time $2^{2^{O(k)}}\poly(n)$. 
+
+\scut{} is easy on trees and the flow-cut gap is 1 for trees. One explaination mentioned in \citep{sparsest_cut_notes} is that shortest path distance in trees is an $\ell_1$ metric. There are works concerning planar graphs and more generally graphs with constant genus.
+\citep{leighton_multicommodity_1999} provided a $\Omega(\log n)$ lowerbound for flow-cut gap for \scut{}. However, it is conjectured that the gap is $O(1)$, while currently the best upperbound is still $O(\sqrt{\log n})$ \citep{rao_small_1999}.
+For graphs with constant genus, \citep{lee_genus_2010} gives a $O(\sqrt{\log g})$ approximation for \scut{}, where $g$ is the genus of the input graph.  For flow-cut gap in planar graphs the techniques are mainly related to metric embedding theory.
 \section{The Research Design}
 % Requirement : Your research design may include exact details of your design and the information should be presented in coherent paragraphs:
 % Example:
@@ -71,12 +78,16 @@ For \nonuscut{} \citep{leighton_multicommodity_1999} only guarantees a $O(\log^2
 % Data analysis (the specific data analysis method)
 % e.g. Using SPSS to analyze the survey data and Using NVivo to analyze the interview data (details of the method and reasons for the choice)
 % The significance/ implications of the study
-  
+\paragraph{Research type:} theoretical research
+\paragraph{Possible difficulties:} The technical depth of open problems in \scut{} might be larger than I expected. If I have no idea after thoroughly understanding metric embedding methods and SDP relaxation, I will immediately move to other problems.
 
 \section{Time Table}
 % Data collection: e.g. During the program and first 6 months after the program (Aug. 2023- May. 2024)
 % Data analysis: June 2024- Sept. 2024
 
+understanding existing methods: 2 weeks.\newline
+solving a problem or imporving some approximation: at most 2 months.
+
 \bibliographystyle{plainnat}
 \bibliography{ref}
 \end{document}

ref.bib

@@ -1,21 +1,4 @@
 
-@article{chalermsook_approximating_2024,
-  title    = {Approximating {Sparsest} {Cut} in {Low}-{Treewidth} {Graphs} via {Combinatorial} {Diameter}},
-  volume   = {20},
-  issn     = {1549-6325, 1549-6333},
-  url      = {http://arxiv.org/abs/2111.06299},
-  doi      = {10.1145/3632623},
-  number   = {1},
-  urldate  = {2025-05-07},
-  journal  = {ACM Transactions on Algorithms},
-  author   = {Chalermsook, Parinya and Kaul, Matthias and Mnich, Matthias and Spoerhase, Joachim and Uniyal, Sumedha and Vaz, Daniel},
-  month    = jan,
-  year     = {2024},
-  note     = {arXiv:2111.06299 [cs]},
-  keywords = {Computer Science - Data Structures and Algorithms},
-  pages    = {1--20},
-  annote   = {Comment: 15 pages, 3 figures}
-}
 @misc{sparsest_cut_notes,
   author       = {Chekuri, Chandra},
   title        = {Introduction to Sparsest Cut},
@@ -42,7 +25,7 @@
 }
 
 @article{arora_osqrtlogn_2010,
-  title    = {\${O}({\textbackslash}sqrt\{{\textbackslash}logn\})\$ {Approximation} to {SPARSEST} {CUT} in \${\textbackslash}tilde\{{O}\}(n{\textasciicircum}2)\$ {Time}},
+  title    = { {$O(\sqrt{\log n})$} {Approximation} to {SPARSEST} {CUT} in {$\tilde{O}(n^2)$} {Time}},
   volume   = {39},
   issn     = {0097-5397, 1095-7111},
   url      = {http://epubs.siam.org/doi/10.1137/080731049},
@@ -187,7 +170,7 @@
 }
 @article{lognGapAumann98,
   author   = {Aumann, Yonatan and Rabani, Yuval},
-  title    = {An O(log k) Approximate Min-Cut Max-Flow Theorem and Approximation Algorithm},
+  title    = {An {$O(\log k)$} Approximate Min-Cut Max-Flow Theorem and Approximation Algorithm},
   journal  = {SIAM Journal on Computing},
   volume   = {27},
   number   = {1},
@@ -198,3 +181,47 @@
   eprint   = {https://doi.org/10.1137/S0097539794285983},
 }
 @article{Linial_London_Rabinovich_1995, title={The geometry of graphs and some of its algorithmic applications}, volume={15}, rights={http://www.springer.com/tdm}, ISSN={0209-9683, 1439-6912}, url={http://link.springer.com/10.1007/BF01200757}, DOI={10.1007/BF01200757}, number={2}, journal={Combinatorica}, author={Linial, Nathan and London, Eran and Rabinovich, Yuri}, year={1995}, month=jun, pages={215–245}, language={en} }
+
+@article{Cohen-Addad_Mömke_Verdugo_2024, title={A 2-approximation for the bounded treewidth sparsest cut problem in $\textsf{FPT}$Time}, volume={206}, ISSN={1436-4646}, url={https://doi.org/10.1007/s10107-023-02044-1}, DOI={10.1007/s10107-023-02044-1}, number={1}, journal={Mathematical Programming}, author={Cohen-Addad, Vincent and Mömke, Tobias and Verdugo, Victor}, year={2024}, month=jul, pages={479–495}, language={en} }
+@INPROCEEDINGS{uniformhardnessFocs07,
+  author={Ambuhl, Christoph and Mastrolilli, Monaldo and Svensson, Ola},
+  booktitle={48th Annual IEEE Symposium on Foundations of Computer Science (FOCS'07)}, 
+  title={Inapproximability Results for Sparsest Cut, Optimal Linear Arrangement, and Precedence Constrained Scheduling}, 
+  year={2007},
+  volume={},
+  number={},
+  pages={329-337},
+  keywords={Cost function;Processor scheduling;Polynomials;NP-complete problem;Approximation algorithms;Single machine scheduling;Computer science},
+  doi={10.1109/FOCS.2007.40}}
+
+@article{bonsma_complexity_2012,
+	title = {The complexity of finding uniform sparsest cuts in various graph classes},
+	volume = {14},
+	copyright = {https://www.elsevier.com/tdm/userlicense/1.0/},
+	issn = {15708667},
+	url = {https://linkinghub.elsevier.com/retrieve/pii/S1570866711001110},
+	doi = {10.1016/j.jda.2011.12.008},
+	language = {en},
+	urldate = {2025-05-10},
+	journal = {Journal of Discrete Algorithms},
+	author = {Bonsma, Paul and Broersma, Hajo and Patel, Viresh and Pyatkin, Artem},
+	month = jul,
+	year = {2012},
+	pages = {136--149},
+}
+
+@inproceedings{rao_small_1999,
+	address = {Miami Beach Florida USA},
+	title = {Small distortion and volume preserving embeddings for planar and {Euclidean} metrics},
+	isbn = {978-1-58113-068-3},
+	url = {https://dl.acm.org/doi/10.1145/304893.304983},
+	doi = {10.1145/304893.304983},
+	language = {en},
+	urldate = {2025-05-10},
+	booktitle = {Proceedings of the fifteenth annual symposium on {Computational} geometry},
+	publisher = {ACM},
+	author = {Rao, Satish},
+	month = jun,
+	year = {1999},
+	pages = {300--306},
+}