From 2cdc315849ac9a079a052b2332a51db537c7e7f0 Mon Sep 17 00:00:00 2001
From: Yu Cong <sxlxcsxlxc@gmail.com>
Date: Sat, 10 May 2025 14:36:22 +0800
Subject: [PATCH] approx rate for general graph sparsest cut

---
 main.pdf | Bin 86993 -> 92644 bytes
 main.tex |   4 +
 ref.bib  | 305 +++++++++++++++++++++++++++++--------------------------
 3 files changed, 164 insertions(+), 145 deletions(-)

diff --git a/main.pdf b/main.pdf
index aed7964b4fad0e96bae36653bed03e84a373f7db..fde243a9667cd3c2e14f15d122b73929145ba207 100644
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diff --git a/main.tex b/main.tex
index 7bb451f..143e447 100644
--- a/main.tex
+++ b/main.tex
@@ -56,7 +56,11 @@ One major open problem for \scut{} is the best approximation ratio for planar gr
 
 \section{Literature Review}
 % Requirement: summarize previous research contributions and identify the gap or the specific problem
+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. 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 it is possible to approximate (non-uniform) \scut{} with better ratio. This paper showed that $O(\log^2 n)$ is the best ratio we can achieve using flow-cut gap.
 
+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{}.
 \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:
diff --git a/ref.bib b/ref.bib
index 377f60b..cfdc7a6 100644
--- a/ref.bib
+++ b/ref.bib
@@ -1,20 +1,20 @@
 
 @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},
+  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},
@@ -26,160 +26,175 @@
 }
 
 @article{hoory_expander_2006,
-	title = {Expander graphs and their applications},
-	volume = {43},
-	issn = {0273-0979},
-	url = {http://www.ams.org/journal-getitem?pii=S0273-0979-06-01126-8},
-	doi = {10.1090/S0273-0979-06-01126-8},
-	language = {en},
-	number = {04},
-	urldate = {2025-05-09},
-	journal = {Bulletin of the American Mathematical Society},
-	author = {Hoory, Shlomo and Linial, Nathan and Wigderson, Avi},
-	month = aug,
-	year = {2006},
-	pages = {439--562},
+  title    = {Expander graphs and their applications},
+  volume   = {43},
+  issn     = {0273-0979},
+  url      = {http://www.ams.org/journal-getitem?pii=S0273-0979-06-01126-8},
+  doi      = {10.1090/S0273-0979-06-01126-8},
+  language = {en},
+  number   = {04},
+  urldate  = {2025-05-09},
+  journal  = {Bulletin of the American Mathematical Society},
+  author   = {Hoory, Shlomo and Linial, Nathan and Wigderson, Avi},
+  month    = aug,
+  year     = {2006},
+  pages    = {439--562}
 }
 
 @article{arora_osqrtlogn_2010,
-	title = {\${O}({\textbackslash}sqrt\{{\textbackslash}logn\})\$ {Approximation} to {SPARSEST} {CUT} in \${\textbackslash}tilde\{{O}\}(n{\textasciicircum}2)\$ {Time}},
-	volume = {39},
-	issn = {0097-5397, 1095-7111},
-	url = {http://epubs.siam.org/doi/10.1137/080731049},
-	doi = {10.1137/080731049},
-	language = {en},
-	number = {5},
-	urldate = {2025-05-09},
-	journal = {SIAM Journal on Computing},
-	author = {Arora, Sanjeev and Hazan, Elad and Kale, Satyen},
-	month = jan,
-	year = {2010},
-	pages = {1748--1771},
+  title    = {\${O}({\textbackslash}sqrt\{{\textbackslash}logn\})\$ {Approximation} to {SPARSEST} {CUT} in \${\textbackslash}tilde\{{O}\}(n{\textasciicircum}2)\$ {Time}},
+  volume   = {39},
+  issn     = {0097-5397, 1095-7111},
+  url      = {http://epubs.siam.org/doi/10.1137/080731049},
+  doi      = {10.1137/080731049},
+  language = {en},
+  number   = {5},
+  urldate  = {2025-05-09},
+  journal  = {SIAM Journal on Computing},
+  author   = {Arora, Sanjeev and Hazan, Elad and Kale, Satyen},
+  month    = jan,
+  year     = {2010},
+  pages    = {1748--1771}
 }
 
 @misc{dorsi2024sparsestcuteigenvaluemultiplicities,
-      title={Sparsest cut and eigenvalue multiplicities on low degree Abelian Cayley graphs}, 
-      author={Tommaso d'Orsi and Chris Jones and Jake Ruotolo and Salil Vadhan and Jiyu Zhang},
-      year={2024},
-      eprint={2412.17115},
-      archivePrefix={arXiv},
-      primaryClass={cs.DS},
-      url={https://arxiv.org/abs/2412.17115}, 
+  title         = {Sparsest cut and eigenvalue multiplicities on low degree Abelian Cayley graphs},
+  author        = {Tommaso d'Orsi and Chris Jones and Jake Ruotolo and Salil Vadhan and Jiyu Zhang},
+  year          = {2024},
+  eprint        = {2412.17115},
+  archiveprefix = {arXiv},
+  primaryclass  = {cs.DS},
+  url           = {https://arxiv.org/abs/2412.17115}
 }
 
 @inproceedings{arora_expander_2004,
-	address = {New York, NY, USA},
-	series = {{STOC} '04},
-	title = {Expander flows, geometric embeddings and graph partitioning},
-	isbn = {978-1-58113-852-8},
-	url = {https://doi.org/10.1145/1007352.1007355},
-	doi = {10.1145/1007352.1007355},
-	abstract = {We give a O(√log n)-approximation algorithm for sparsest cut, balanced separator, and graph conductance problems. This improves the O(log n)-approximation of Leighton and Rao (1988). We use a well-known semidefinite relaxation with triangle inequality constraints. Central to our analysis is a geometric theorem about projections of point sets in Rd, whose proof makes essential use of a phenomenon called measure concentration. We also describe an interesting and natural "certificate" for a graph's expansion, by embedding an n-node expander in it with appropriate dilation and congestion. We call this an expander flow.},
-	urldate = {2025-05-09},
-	booktitle = {Proceedings of the thirty-sixth annual {ACM} symposium on {Theory} of computing},
-	publisher = {Association for Computing Machinery},
-	author = {Arora, Sanjeev and Rao, Satish and Vazirani, Umesh},
-	month = jun,
-	year = {2004},
-	pages = {222--231},
+  address   = {New York, NY, USA},
+  series    = {{STOC} '04},
+  title     = {Expander flows, geometric embeddings and graph partitioning},
+  isbn      = {978-1-58113-852-8},
+  url       = {https://doi.org/10.1145/1007352.1007355},
+  doi       = {10.1145/1007352.1007355},
+  abstract  = {We give a O(√log n)-approximation algorithm for sparsest cut, balanced separator, and graph conductance problems. This improves the O(log n)-approximation of Leighton and Rao (1988). We use a well-known semidefinite relaxation with triangle inequality constraints. Central to our analysis is a geometric theorem about projections of point sets in Rd, whose proof makes essential use of a phenomenon called measure concentration. We also describe an interesting and natural "certificate" for a graph's expansion, by embedding an n-node expander in it with appropriate dilation and congestion. We call this an expander flow.},
+  urldate   = {2025-05-09},
+  booktitle = {Proceedings of the thirty-sixth annual {ACM} symposium on {Theory} of computing},
+  publisher = {Association for Computing Machinery},
+  author    = {Arora, Sanjeev and Rao, Satish and Vazirani, Umesh},
+  month     = jun,
+  year      = {2004},
+  pages     = {222--231}
 }
 
 @article{leighton_multicommodity_1999,
-	title = {Multicommodity max-flow min-cut theorems and their use in designing approximation algorithms},
-	volume = {46},
-	issn = {0004-5411},
-	url = {https://dl.acm.org/doi/10.1145/331524.331526},
-	doi = {10.1145/331524.331526},
-	number = {6},
-	urldate = {2025-05-09},
-	journal = {J. ACM},
-	author = {Leighton, Tom and Rao, Satish},
-	month = nov,
-	year = {1999},
-	pages = {787--832},
+  title   = {Multicommodity max-flow min-cut theorems and their use in designing approximation algorithms},
+  volume  = {46},
+  issn    = {0004-5411},
+  url     = {https://dl.acm.org/doi/10.1145/331524.331526},
+  doi     = {10.1145/331524.331526},
+  number  = {6},
+  urldate = {2025-05-09},
+  journal = {J. ACM},
+  author  = {Leighton, Tom and Rao, Satish},
+  month   = nov,
+  year    = {1999},
+  pages   = {787--832}
 }
 
 @inproceedings{lee_genus_2010,
-	title = {Genus and the geometry of the cut graph: [extended abstract]},
-	isbn = {978-0-89871-701-3 978-1-61197-307-5},
-	shorttitle = {Genus and the geometry of the cut graph},
-	url = {https://epubs.siam.org/doi/10.1137/1.9781611973075.18},
-	doi = {10.1137/1.9781611973075.18},
-	language = {en},
-	urldate = {2025-05-07},
-	booktitle = {Proceedings of the {Twenty}-{First} {Annual} {ACM}-{SIAM} {Symposium} on {Discrete} {Algorithms}},
-	publisher = {Society for Industrial and Applied Mathematics},
-	author = {Lee, James R. and Sidiropoulos, Anastasios},
-	month = jan,
-	year = {2010},
-	pages = {193--201},
+  title      = {Genus and the geometry of the cut graph: [extended abstract]},
+  isbn       = {978-0-89871-701-3 978-1-61197-307-5},
+  shorttitle = {Genus and the geometry of the cut graph},
+  url        = {https://epubs.siam.org/doi/10.1137/1.9781611973075.18},
+  doi        = {10.1137/1.9781611973075.18},
+  language   = {en},
+  urldate    = {2025-05-07},
+  booktitle  = {Proceedings of the {Twenty}-{First} {Annual} {ACM}-{SIAM} {Symposium} on {Discrete} {Algorithms}},
+  publisher  = {Society for Industrial and Applied Mathematics},
+  author     = {Lee, James R. and Sidiropoulos, Anastasios},
+  month      = jan,
+  year       = {2010},
+  pages      = {193--201}
 }
 @misc{gupta2013sparsestcutboundedtreewidth,
-      title={Sparsest Cut on Bounded Treewidth Graphs: Algorithms and Hardness Results}, 
-      author={Anupam Gupta and Kunal Talwar and David Witmer},
-      year={2013},
-      eprint={1305.1347},
-      archivePrefix={arXiv},
-      primaryClass={cs.DS},
-      url={https://arxiv.org/abs/1305.1347}, 
+  title         = {Sparsest Cut on Bounded Treewidth Graphs: Algorithms and Hardness Results},
+  author        = {Anupam Gupta and Kunal Talwar and David Witmer},
+  year          = {2013},
+  eprint        = {1305.1347},
+  archiveprefix = {arXiv},
+  primaryclass  = {cs.DS},
+  url           = {https://arxiv.org/abs/1305.1347}
 }
 @article{Chalermsook_2024,
-   title={Approximating Sparsest Cut in Low-treewidth Graphs via Combinatorial Diameter},
-   volume={20},
-   ISSN={1549-6333},
-   url={http://dx.doi.org/10.1145/3632623},
-   DOI={10.1145/3632623},
-   number={1},
-   journal={ACM Transactions on Algorithms},
-   publisher={Association for Computing Machinery (ACM)},
-   author={Chalermsook, Parinya and Kaul, Matthias and Mnich, Matthias and Spoerhase, Joachim and Uniyal, Sumedha and Vaz, Daniel},
-   year={2024},
-   month=jan, pages={1–20} }
+  title     = {Approximating Sparsest Cut in Low-treewidth Graphs via Combinatorial Diameter},
+  volume    = {20},
+  issn      = {1549-6333},
+  url       = {http://dx.doi.org/10.1145/3632623},
+  doi       = {10.1145/3632623},
+  number    = {1},
+  journal   = {ACM Transactions on Algorithms},
+  publisher = {Association for Computing Machinery (ACM)},
+  author    = {Chalermsook, Parinya and Kaul, Matthias and Mnich, Matthias and Spoerhase, Joachim and Uniyal, Sumedha and Vaz, Daniel},
+  year      = {2024},
+  month     = jan,
+  pages     = {1–20}
+}
 @article{juliaJACMapxhard,
-author = {Chuzhoy, Julia and Khanna, Sanjeev},
-title = {Polynomial flow-cut gaps and hardness of directed cut problems},
-year = {2009},
-issue_date = {April 2009},
-publisher = {Association for Computing Machinery},
-address = {New York, NY, USA},
-volume = {56},
-number = {2},
-issn = {0004-5411},
-url = {https://doi.org/10.1145/1502793.1502795},
-doi = {10.1145/1502793.1502795},
-journal = {J. ACM},
-month = apr,
-articleno = {6},
-numpages = {28},
-keywords = {sparsest cut, hardness of approximation, Directed multicut}
+  author     = {Chuzhoy, Julia and Khanna, Sanjeev},
+  title      = {Polynomial flow-cut gaps and hardness of directed cut problems},
+  year       = {2009},
+  issue_date = {April 2009},
+  publisher  = {Association for Computing Machinery},
+  address    = {New York, NY, USA},
+  volume     = {56},
+  number     = {2},
+  issn       = {0004-5411},
+  url        = {https://doi.org/10.1145/1502793.1502795},
+  doi        = {10.1145/1502793.1502795},
+  journal    = {J. ACM},
+  month      = apr,
+  articleno  = {6},
+  numpages   = {28},
+  keywords   = {sparsest cut, hardness of approximation, Directed multicut}
 }
 
 @inproceedings{chawla_hardness_2005,
-	title = {On the hardness of approximating {MULTICUT} and {SPARSEST}-{CUT}},
-	url = {https://ieeexplore.ieee.org/document/1443081},
-	doi = {10.1109/CCC.2005.20},
-	abstract = {We show that the MULTICUT, SPARSEST-CUT, and MIN-2CNF/spl equiv/DELETION problems are NP-hard to approximate within every constant factor, assuming the unique games conjecture of Khot [STOC, 2002]. A quantitatively stronger version of the conjecture implies inapproximability factor of /spl Omega/(log log n).},
-	urldate = {2025-05-09},
-	booktitle = {20th {Annual} {IEEE} {Conference} on {Computational} {Complexity} ({CCC}'05)},
-	author = {Chawla, S. and Krauthgamer, R. and Kumar, R. and Rabani, Y. and Sivakumar, D.},
-	month = jun,
-	year = {2005},
-	note = {ISSN: 1093-0159},
-	keywords = {Approximation algorithms, Computer science, Costs, Linear programming, Mathematics},
-	pages = {144--153},
+  title     = {On the hardness of approximating {MULTICUT} and {SPARSEST}-{CUT}},
+  url       = {https://ieeexplore.ieee.org/document/1443081},
+  doi       = {10.1109/CCC.2005.20},
+  abstract  = {We show that the MULTICUT, SPARSEST-CUT, and MIN-2CNF/spl equiv/DELETION problems are NP-hard to approximate within every constant factor, assuming the unique games conjecture of Khot [STOC, 2002]. A quantitatively stronger version of the conjecture implies inapproximability factor of /spl Omega/(log log n).},
+  urldate   = {2025-05-09},
+  booktitle = {20th {Annual} {IEEE} {Conference} on {Computational} {Complexity} ({CCC}'05)},
+  author    = {Chawla, S. and Krauthgamer, R. and Kumar, R. and Rabani, Y. and Sivakumar, D.},
+  month     = jun,
+  year      = {2005},
+  note      = {ISSN: 1093-0159},
+  keywords  = {Approximation algorithms, Computer science, Costs, Linear programming, Mathematics},
+  pages     = {144--153}
 }
 
 @inproceedings{chlamtac_approximating_2010,
-	address = {Berlin, Heidelberg},
-	title = {Approximating {Sparsest} {Cut} in {Graphs} of {Bounded} {Treewidth}},
-	isbn = {978-3-642-15369-3},
-	doi = {10.1007/978-3-642-15369-3_10},
-	language = {en},
-	booktitle = {Approximation, {Randomization}, and {Combinatorial} {Optimization}. {Algorithms} and {Techniques}},
-	publisher = {Springer},
-	author = {Chlamtac, Eden and Krauthgamer, Robert and Raghavendra, Prasad},
-	editor = {Serna, Maria and Shaltiel, Ronen and Jansen, Klaus and Rolim, José},
-	year = {2010},
-	keywords = {General Demand, Linear Programming Relaxation, Linear Programming Solution, Tree Decomposition, Vertex Cover},
-	pages = {124--137},
+  address   = {Berlin, Heidelberg},
+  title     = {Approximating {Sparsest} {Cut} in {Graphs} of {Bounded} {Treewidth}},
+  isbn      = {978-3-642-15369-3},
+  doi       = {10.1007/978-3-642-15369-3_10},
+  language  = {en},
+  booktitle = {Approximation, {Randomization}, and {Combinatorial} {Optimization}. {Algorithms} and {Techniques}},
+  publisher = {Springer},
+  author    = {Chlamtac, Eden and Krauthgamer, Robert and Raghavendra, Prasad},
+  editor    = {Serna, Maria and Shaltiel, Ronen and Jansen, Klaus and Rolim, José},
+  year      = {2010},
+  keywords  = {General Demand, Linear Programming Relaxation, Linear Programming Solution, Tree Decomposition, Vertex Cover},
+  pages     = {124--137}
 }
+@article{lognGapAumann98,
+  author   = {Aumann, Yonatan and Rabani, Yuval},
+  title    = {An O(log k) Approximate Min-Cut Max-Flow Theorem and Approximation Algorithm},
+  journal  = {SIAM Journal on Computing},
+  volume   = {27},
+  number   = {1},
+  pages    = {291-301},
+  year     = {1998},
+  doi      = {10.1137/S0097539794285983},
+  url      = {https://doi.org/10.1137/S0097539794285983},
+  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} }