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approx rate for general graph sparsest cut

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@ -56,7 +56,11 @@ One major open problem for \scut{} is the best approximation ratio for planar gr
\section{Literature Review} \section{Literature Review}
% Requirement: summarize previous research contributions and identify the gap or the specific problem % 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} \section{The Research Design}
% Requirement : Your research design may include exact details of your design and the information should be presented in coherent paragraphs: % Requirement : Your research design may include exact details of your design and the information should be presented in coherent paragraphs:
% Example: % Example:

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ref.bib
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@ -14,7 +14,7 @@
note = {arXiv:2111.06299 [cs]}, note = {arXiv:2111.06299 [cs]},
keywords = {Computer Science - Data Structures and Algorithms}, keywords = {Computer Science - Data Structures and Algorithms},
pages = {1--20}, pages = {1--20},
annote = {Comment: 15 pages, 3 figures}, annote = {Comment: 15 pages, 3 figures}
} }
@misc{sparsest_cut_notes, @misc{sparsest_cut_notes,
author = {Chekuri, Chandra}, author = {Chekuri, Chandra},
@ -38,7 +38,7 @@
author = {Hoory, Shlomo and Linial, Nathan and Wigderson, Avi}, author = {Hoory, Shlomo and Linial, Nathan and Wigderson, Avi},
month = aug, month = aug,
year = {2006}, year = {2006},
pages = {439--562}, pages = {439--562}
} }
@article{arora_osqrtlogn_2010, @article{arora_osqrtlogn_2010,
@ -54,17 +54,17 @@
author = {Arora, Sanjeev and Hazan, Elad and Kale, Satyen}, author = {Arora, Sanjeev and Hazan, Elad and Kale, Satyen},
month = jan, month = jan,
year = {2010}, year = {2010},
pages = {1748--1771}, pages = {1748--1771}
} }
@misc{dorsi2024sparsestcuteigenvaluemultiplicities, @misc{dorsi2024sparsestcuteigenvaluemultiplicities,
title={Sparsest cut and eigenvalue multiplicities on low degree Abelian Cayley graphs}, 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}, author = {Tommaso d'Orsi and Chris Jones and Jake Ruotolo and Salil Vadhan and Jiyu Zhang},
year={2024}, year = {2024},
eprint={2412.17115}, eprint = {2412.17115},
archivePrefix={arXiv}, archiveprefix = {arXiv},
primaryClass={cs.DS}, primaryclass = {cs.DS},
url={https://arxiv.org/abs/2412.17115}, url = {https://arxiv.org/abs/2412.17115}
} }
@inproceedings{arora_expander_2004, @inproceedings{arora_expander_2004,
@ -81,7 +81,7 @@
author = {Arora, Sanjeev and Rao, Satish and Vazirani, Umesh}, author = {Arora, Sanjeev and Rao, Satish and Vazirani, Umesh},
month = jun, month = jun,
year = {2004}, year = {2004},
pages = {222--231}, pages = {222--231}
} }
@article{leighton_multicommodity_1999, @article{leighton_multicommodity_1999,
@ -96,7 +96,7 @@
author = {Leighton, Tom and Rao, Satish}, author = {Leighton, Tom and Rao, Satish},
month = nov, month = nov,
year = {1999}, year = {1999},
pages = {787--832}, pages = {787--832}
} }
@inproceedings{lee_genus_2010, @inproceedings{lee_genus_2010,
@ -112,46 +112,48 @@
author = {Lee, James R. and Sidiropoulos, Anastasios}, author = {Lee, James R. and Sidiropoulos, Anastasios},
month = jan, month = jan,
year = {2010}, year = {2010},
pages = {193--201}, pages = {193--201}
} }
@misc{gupta2013sparsestcutboundedtreewidth, @misc{gupta2013sparsestcutboundedtreewidth,
title={Sparsest Cut on Bounded Treewidth Graphs: Algorithms and Hardness Results}, title = {Sparsest Cut on Bounded Treewidth Graphs: Algorithms and Hardness Results},
author={Anupam Gupta and Kunal Talwar and David Witmer}, author = {Anupam Gupta and Kunal Talwar and David Witmer},
year={2013}, year = {2013},
eprint={1305.1347}, eprint = {1305.1347},
archivePrefix={arXiv}, archiveprefix = {arXiv},
primaryClass={cs.DS}, primaryclass = {cs.DS},
url={https://arxiv.org/abs/1305.1347}, url = {https://arxiv.org/abs/1305.1347}
} }
@article{Chalermsook_2024, @article{Chalermsook_2024,
title={Approximating Sparsest Cut in Low-treewidth Graphs via Combinatorial Diameter}, title = {Approximating Sparsest Cut in Low-treewidth Graphs via Combinatorial Diameter},
volume={20}, volume = {20},
ISSN={1549-6333}, issn = {1549-6333},
url={http://dx.doi.org/10.1145/3632623}, url = {http://dx.doi.org/10.1145/3632623},
DOI={10.1145/3632623}, doi = {10.1145/3632623},
number={1}, number = {1},
journal={ACM Transactions on Algorithms}, journal = {ACM Transactions on Algorithms},
publisher={Association for Computing Machinery (ACM)}, publisher = {Association for Computing Machinery (ACM)},
author={Chalermsook, Parinya and Kaul, Matthias and Mnich, Matthias and Spoerhase, Joachim and Uniyal, Sumedha and Vaz, Daniel}, author = {Chalermsook, Parinya and Kaul, Matthias and Mnich, Matthias and Spoerhase, Joachim and Uniyal, Sumedha and Vaz, Daniel},
year={2024}, year = {2024},
month=jan, pages={1–20} } month = jan,
pages = {1–20}
}
@article{juliaJACMapxhard, @article{juliaJACMapxhard,
author = {Chuzhoy, Julia and Khanna, Sanjeev}, author = {Chuzhoy, Julia and Khanna, Sanjeev},
title = {Polynomial flow-cut gaps and hardness of directed cut problems}, title = {Polynomial flow-cut gaps and hardness of directed cut problems},
year = {2009}, year = {2009},
issue_date = {April 2009}, issue_date = {April 2009},
publisher = {Association for Computing Machinery}, publisher = {Association for Computing Machinery},
address = {New York, NY, USA}, address = {New York, NY, USA},
volume = {56}, volume = {56},
number = {2}, number = {2},
issn = {0004-5411}, issn = {0004-5411},
url = {https://doi.org/10.1145/1502793.1502795}, url = {https://doi.org/10.1145/1502793.1502795},
doi = {10.1145/1502793.1502795}, doi = {10.1145/1502793.1502795},
journal = {J. ACM}, journal = {J. ACM},
month = apr, month = apr,
articleno = {6}, articleno = {6},
numpages = {28}, numpages = {28},
keywords = {sparsest cut, hardness of approximation, Directed multicut} keywords = {sparsest cut, hardness of approximation, Directed multicut}
} }
@inproceedings{chawla_hardness_2005, @inproceedings{chawla_hardness_2005,
@ -166,7 +168,7 @@ keywords = {sparsest cut, hardness of approximation, Directed multicut}
year = {2005}, year = {2005},
note = {ISSN: 1093-0159}, note = {ISSN: 1093-0159},
keywords = {Approximation algorithms, Computer science, Costs, Linear programming, Mathematics}, keywords = {Approximation algorithms, Computer science, Costs, Linear programming, Mathematics},
pages = {144--153}, pages = {144--153}
} }
@inproceedings{chlamtac_approximating_2010, @inproceedings{chlamtac_approximating_2010,
@ -181,5 +183,18 @@ keywords = {sparsest cut, hardness of approximation, Directed multicut}
editor = {Serna, Maria and Shaltiel, Ronen and Jansen, Klaus and Rolim, José}, editor = {Serna, Maria and Shaltiel, Ronen and Jansen, Klaus and Rolim, José},
year = {2010}, year = {2010},
keywords = {General Demand, Linear Programming Relaxation, Linear Programming Solution, Tree Decomposition, Vertex Cover}, keywords = {General Demand, Linear Programming Relaxation, Linear Programming Solution, Tree Decomposition, Vertex Cover},
pages = {124--137}, 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} }