approx rate for general graph sparsest cut
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main.tex
@ -56,7 +56,11 @@ One major open problem for \scut{} is the best approximation ratio for planar gr
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\section{Literature Review}
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% Requirement: summarize previous research contributions and identify the gap or the specific problem
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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.
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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.
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\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{}.
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\section{The Research Design}
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% Requirement : Your research design may include exact details of your design and the information should be presented in coherent paragraphs:
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% Example:
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49
ref.bib
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ref.bib
@ -14,7 +14,7 @@
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note = {arXiv:2111.06299 [cs]},
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keywords = {Computer Science - Data Structures and Algorithms},
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pages = {1--20},
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annote = {Comment: 15 pages, 3 figures},
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annote = {Comment: 15 pages, 3 figures}
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}
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@misc{sparsest_cut_notes,
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author = {Chekuri, Chandra},
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@ -38,7 +38,7 @@
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author = {Hoory, Shlomo and Linial, Nathan and Wigderson, Avi},
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month = aug,
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year = {2006},
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pages = {439--562},
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pages = {439--562}
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}
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@article{arora_osqrtlogn_2010,
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@ -54,7 +54,7 @@
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author = {Arora, Sanjeev and Hazan, Elad and Kale, Satyen},
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month = jan,
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year = {2010},
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pages = {1748--1771},
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pages = {1748--1771}
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}
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@misc{dorsi2024sparsestcuteigenvaluemultiplicities,
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@ -62,9 +62,9 @@
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author = {Tommaso d'Orsi and Chris Jones and Jake Ruotolo and Salil Vadhan and Jiyu Zhang},
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year = {2024},
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eprint = {2412.17115},
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archivePrefix={arXiv},
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primaryClass={cs.DS},
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url={https://arxiv.org/abs/2412.17115},
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archiveprefix = {arXiv},
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primaryclass = {cs.DS},
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url = {https://arxiv.org/abs/2412.17115}
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}
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@inproceedings{arora_expander_2004,
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@ -81,7 +81,7 @@
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author = {Arora, Sanjeev and Rao, Satish and Vazirani, Umesh},
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month = jun,
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year = {2004},
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pages = {222--231},
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pages = {222--231}
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}
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@article{leighton_multicommodity_1999,
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@ -96,7 +96,7 @@
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author = {Leighton, Tom and Rao, Satish},
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month = nov,
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year = {1999},
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pages = {787--832},
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pages = {787--832}
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}
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@inproceedings{lee_genus_2010,
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@ -112,29 +112,31 @@
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author = {Lee, James R. and Sidiropoulos, Anastasios},
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month = jan,
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year = {2010},
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pages = {193--201},
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pages = {193--201}
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}
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@misc{gupta2013sparsestcutboundedtreewidth,
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title = {Sparsest Cut on Bounded Treewidth Graphs: Algorithms and Hardness Results},
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author = {Anupam Gupta and Kunal Talwar and David Witmer},
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year = {2013},
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eprint = {1305.1347},
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archivePrefix={arXiv},
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primaryClass={cs.DS},
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url={https://arxiv.org/abs/1305.1347},
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archiveprefix = {arXiv},
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primaryclass = {cs.DS},
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url = {https://arxiv.org/abs/1305.1347}
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}
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@article{Chalermsook_2024,
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title = {Approximating Sparsest Cut in Low-treewidth Graphs via Combinatorial Diameter},
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volume = {20},
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ISSN={1549-6333},
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issn = {1549-6333},
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url = {http://dx.doi.org/10.1145/3632623},
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DOI={10.1145/3632623},
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doi = {10.1145/3632623},
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number = {1},
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journal = {ACM Transactions on Algorithms},
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publisher = {Association for Computing Machinery (ACM)},
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author = {Chalermsook, Parinya and Kaul, Matthias and Mnich, Matthias and Spoerhase, Joachim and Uniyal, Sumedha and Vaz, Daniel},
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year = {2024},
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month=jan, pages={1–20} }
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month = jan,
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pages = {1–20}
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}
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@article{juliaJACMapxhard,
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author = {Chuzhoy, Julia and Khanna, Sanjeev},
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title = {Polynomial flow-cut gaps and hardness of directed cut problems},
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@ -166,7 +168,7 @@ keywords = {sparsest cut, hardness of approximation, Directed multicut}
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year = {2005},
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note = {ISSN: 1093-0159},
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keywords = {Approximation algorithms, Computer science, Costs, Linear programming, Mathematics},
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pages = {144--153},
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pages = {144--153}
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}
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@inproceedings{chlamtac_approximating_2010,
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@ -181,5 +183,18 @@ keywords = {sparsest cut, hardness of approximation, Directed multicut}
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editor = {Serna, Maria and Shaltiel, Ronen and Jansen, Klaus and Rolim, José},
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year = {2010},
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keywords = {General Demand, Linear Programming Relaxation, Linear Programming Solution, Tree Decomposition, Vertex Cover},
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pages = {124--137},
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pages = {124--137}
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}
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@article{lognGapAumann98,
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author = {Aumann, Yonatan and Rabani, Yuval},
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title = {An O(log k) Approximate Min-Cut Max-Flow Theorem and Approximation Algorithm},
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journal = {SIAM Journal on Computing},
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volume = {27},
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number = {1},
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pages = {291-301},
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year = {1998},
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doi = {10.1137/S0097539794285983},
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url = {https://doi.org/10.1137/S0097539794285983},
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eprint = {https://doi.org/10.1137/S0097539794285983},
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}
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@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} }
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