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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\section{Literature Review}
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% Requirement: summarize previous research contributions and identify the gap or the specific problem
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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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\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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% 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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% Example:
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ref.bib
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ref.bib
@ -1,20 +1,20 @@
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@article{chalermsook_approximating_2024,
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@article{chalermsook_approximating_2024,
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title = {Approximating {Sparsest} {Cut} in {Low}-{Treewidth} {Graphs} via {Combinatorial} {Diameter}},
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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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volume = {20},
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issn = {1549-6325, 1549-6333},
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issn = {1549-6325, 1549-6333},
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url = {http://arxiv.org/abs/2111.06299},
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url = {http://arxiv.org/abs/2111.06299},
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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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number = {1},
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urldate = {2025-05-07},
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urldate = {2025-05-07},
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journal = {ACM Transactions on Algorithms},
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journal = {ACM Transactions on Algorithms},
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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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author = {Chalermsook, Parinya and Kaul, Matthias and Mnich, Matthias and Spoerhase, Joachim and Uniyal, Sumedha and Vaz, Daniel},
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month = jan,
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month = jan,
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year = {2024},
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year = {2024},
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note = {arXiv:2111.06299 [cs]},
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note = {arXiv:2111.06299 [cs]},
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keywords = {Computer Science - Data Structures and Algorithms},
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keywords = {Computer Science - Data Structures and Algorithms},
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pages = {1--20},
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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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}
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@misc{sparsest_cut_notes,
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@misc{sparsest_cut_notes,
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author = {Chekuri, Chandra},
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author = {Chekuri, Chandra},
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@ -26,160 +26,175 @@
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}
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}
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@article{hoory_expander_2006,
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@article{hoory_expander_2006,
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title = {Expander graphs and their applications},
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title = {Expander graphs and their applications},
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volume = {43},
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volume = {43},
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issn = {0273-0979},
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issn = {0273-0979},
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url = {http://www.ams.org/journal-getitem?pii=S0273-0979-06-01126-8},
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url = {http://www.ams.org/journal-getitem?pii=S0273-0979-06-01126-8},
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doi = {10.1090/S0273-0979-06-01126-8},
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doi = {10.1090/S0273-0979-06-01126-8},
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language = {en},
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language = {en},
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number = {04},
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number = {04},
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urldate = {2025-05-09},
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urldate = {2025-05-09},
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journal = {Bulletin of the American Mathematical Society},
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journal = {Bulletin of the American Mathematical Society},
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author = {Hoory, Shlomo and Linial, Nathan and Wigderson, Avi},
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author = {Hoory, Shlomo and Linial, Nathan and Wigderson, Avi},
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month = aug,
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month = aug,
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year = {2006},
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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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}
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@article{arora_osqrtlogn_2010,
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@article{arora_osqrtlogn_2010,
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title = {\${O}({\textbackslash}sqrt\{{\textbackslash}logn\})\$ {Approximation} to {SPARSEST} {CUT} in \${\textbackslash}tilde\{{O}\}(n{\textasciicircum}2)\$ {Time}},
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title = {\${O}({\textbackslash}sqrt\{{\textbackslash}logn\})\$ {Approximation} to {SPARSEST} {CUT} in \${\textbackslash}tilde\{{O}\}(n{\textasciicircum}2)\$ {Time}},
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volume = {39},
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volume = {39},
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issn = {0097-5397, 1095-7111},
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issn = {0097-5397, 1095-7111},
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url = {http://epubs.siam.org/doi/10.1137/080731049},
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url = {http://epubs.siam.org/doi/10.1137/080731049},
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doi = {10.1137/080731049},
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doi = {10.1137/080731049},
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language = {en},
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language = {en},
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number = {5},
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number = {5},
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urldate = {2025-05-09},
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urldate = {2025-05-09},
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journal = {SIAM Journal on Computing},
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journal = {SIAM Journal on Computing},
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author = {Arora, Sanjeev and Hazan, Elad and Kale, Satyen},
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author = {Arora, Sanjeev and Hazan, Elad and Kale, Satyen},
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month = jan,
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month = jan,
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year = {2010},
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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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}
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@misc{dorsi2024sparsestcuteigenvaluemultiplicities,
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@misc{dorsi2024sparsestcuteigenvaluemultiplicities,
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title={Sparsest cut and eigenvalue multiplicities on low degree Abelian Cayley graphs},
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title = {Sparsest cut and eigenvalue multiplicities on low degree Abelian Cayley graphs},
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author={Tommaso d'Orsi and Chris Jones and Jake Ruotolo and Salil Vadhan and Jiyu Zhang},
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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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year = {2024},
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eprint={2412.17115},
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eprint = {2412.17115},
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archivePrefix={arXiv},
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archiveprefix = {arXiv},
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primaryClass={cs.DS},
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primaryclass = {cs.DS},
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url={https://arxiv.org/abs/2412.17115},
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url = {https://arxiv.org/abs/2412.17115}
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}
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}
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@inproceedings{arora_expander_2004,
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@inproceedings{arora_expander_2004,
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address = {New York, NY, USA},
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address = {New York, NY, USA},
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series = {{STOC} '04},
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series = {{STOC} '04},
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title = {Expander flows, geometric embeddings and graph partitioning},
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title = {Expander flows, geometric embeddings and graph partitioning},
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isbn = {978-1-58113-852-8},
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isbn = {978-1-58113-852-8},
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url = {https://doi.org/10.1145/1007352.1007355},
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url = {https://doi.org/10.1145/1007352.1007355},
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doi = {10.1145/1007352.1007355},
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doi = {10.1145/1007352.1007355},
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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.},
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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.},
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urldate = {2025-05-09},
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urldate = {2025-05-09},
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booktitle = {Proceedings of the thirty-sixth annual {ACM} symposium on {Theory} of computing},
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booktitle = {Proceedings of the thirty-sixth annual {ACM} symposium on {Theory} of computing},
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publisher = {Association for Computing Machinery},
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publisher = {Association for Computing Machinery},
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author = {Arora, Sanjeev and Rao, Satish and Vazirani, Umesh},
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author = {Arora, Sanjeev and Rao, Satish and Vazirani, Umesh},
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month = jun,
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month = jun,
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year = {2004},
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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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}
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@article{leighton_multicommodity_1999,
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@article{leighton_multicommodity_1999,
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title = {Multicommodity max-flow min-cut theorems and their use in designing approximation algorithms},
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title = {Multicommodity max-flow min-cut theorems and their use in designing approximation algorithms},
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volume = {46},
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volume = {46},
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issn = {0004-5411},
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issn = {0004-5411},
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url = {https://dl.acm.org/doi/10.1145/331524.331526},
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url = {https://dl.acm.org/doi/10.1145/331524.331526},
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doi = {10.1145/331524.331526},
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doi = {10.1145/331524.331526},
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number = {6},
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number = {6},
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urldate = {2025-05-09},
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urldate = {2025-05-09},
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journal = {J. ACM},
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journal = {J. ACM},
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author = {Leighton, Tom and Rao, Satish},
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author = {Leighton, Tom and Rao, Satish},
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month = nov,
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month = nov,
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year = {1999},
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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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}
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@inproceedings{lee_genus_2010,
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@inproceedings{lee_genus_2010,
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title = {Genus and the geometry of the cut graph: [extended abstract]},
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title = {Genus and the geometry of the cut graph: [extended abstract]},
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isbn = {978-0-89871-701-3 978-1-61197-307-5},
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isbn = {978-0-89871-701-3 978-1-61197-307-5},
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shorttitle = {Genus and the geometry of the cut graph},
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shorttitle = {Genus and the geometry of the cut graph},
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url = {https://epubs.siam.org/doi/10.1137/1.9781611973075.18},
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url = {https://epubs.siam.org/doi/10.1137/1.9781611973075.18},
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doi = {10.1137/1.9781611973075.18},
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doi = {10.1137/1.9781611973075.18},
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language = {en},
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language = {en},
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urldate = {2025-05-07},
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urldate = {2025-05-07},
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booktitle = {Proceedings of the {Twenty}-{First} {Annual} {ACM}-{SIAM} {Symposium} on {Discrete} {Algorithms}},
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booktitle = {Proceedings of the {Twenty}-{First} {Annual} {ACM}-{SIAM} {Symposium} on {Discrete} {Algorithms}},
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publisher = {Society for Industrial and Applied Mathematics},
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publisher = {Society for Industrial and Applied Mathematics},
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author = {Lee, James R. and Sidiropoulos, Anastasios},
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author = {Lee, James R. and Sidiropoulos, Anastasios},
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month = jan,
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month = jan,
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year = {2010},
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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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}
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@misc{gupta2013sparsestcutboundedtreewidth,
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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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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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author = {Anupam Gupta and Kunal Talwar and David Witmer},
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year={2013},
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year = {2013},
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eprint={1305.1347},
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eprint = {1305.1347},
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archivePrefix={arXiv},
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archiveprefix = {arXiv},
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primaryClass={cs.DS},
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primaryclass = {cs.DS},
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url={https://arxiv.org/abs/1305.1347},
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url = {https://arxiv.org/abs/1305.1347}
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}
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}
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@article{Chalermsook_2024,
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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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title = {Approximating Sparsest Cut in Low-treewidth Graphs via Combinatorial Diameter},
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volume={20},
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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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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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number = {1},
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journal={ACM Transactions on Algorithms},
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journal = {ACM Transactions on Algorithms},
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publisher={Association for Computing Machinery (ACM)},
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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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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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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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@article{juliaJACMapxhard,
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author = {Chuzhoy, Julia and Khanna, Sanjeev},
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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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title = {Polynomial flow-cut gaps and hardness of directed cut problems},
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year = {2009},
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year = {2009},
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issue_date = {April 2009},
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issue_date = {April 2009},
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publisher = {Association for Computing Machinery},
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publisher = {Association for Computing Machinery},
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address = {New York, NY, USA},
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address = {New York, NY, USA},
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volume = {56},
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volume = {56},
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number = {2},
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number = {2},
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issn = {0004-5411},
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issn = {0004-5411},
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url = {https://doi.org/10.1145/1502793.1502795},
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url = {https://doi.org/10.1145/1502793.1502795},
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doi = {10.1145/1502793.1502795},
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doi = {10.1145/1502793.1502795},
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journal = {J. ACM},
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journal = {J. ACM},
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month = apr,
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month = apr,
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articleno = {6},
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articleno = {6},
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numpages = {28},
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numpages = {28},
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keywords = {sparsest cut, hardness of approximation, Directed multicut}
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keywords = {sparsest cut, hardness of approximation, Directed multicut}
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}
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}
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@inproceedings{chawla_hardness_2005,
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@inproceedings{chawla_hardness_2005,
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title = {On the hardness of approximating {MULTICUT} and {SPARSEST}-{CUT}},
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title = {On the hardness of approximating {MULTICUT} and {SPARSEST}-{CUT}},
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url = {https://ieeexplore.ieee.org/document/1443081},
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url = {https://ieeexplore.ieee.org/document/1443081},
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doi = {10.1109/CCC.2005.20},
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doi = {10.1109/CCC.2005.20},
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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).},
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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).},
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urldate = {2025-05-09},
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urldate = {2025-05-09},
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booktitle = {20th {Annual} {IEEE} {Conference} on {Computational} {Complexity} ({CCC}'05)},
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booktitle = {20th {Annual} {IEEE} {Conference} on {Computational} {Complexity} ({CCC}'05)},
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author = {Chawla, S. and Krauthgamer, R. and Kumar, R. and Rabani, Y. and Sivakumar, D.},
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author = {Chawla, S. and Krauthgamer, R. and Kumar, R. and Rabani, Y. and Sivakumar, D.},
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month = jun,
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month = jun,
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year = {2005},
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year = {2005},
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note = {ISSN: 1093-0159},
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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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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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}
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@inproceedings{chlamtac_approximating_2010,
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@inproceedings{chlamtac_approximating_2010,
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address = {Berlin, Heidelberg},
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address = {Berlin, Heidelberg},
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title = {Approximating {Sparsest} {Cut} in {Graphs} of {Bounded} {Treewidth}},
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title = {Approximating {Sparsest} {Cut} in {Graphs} of {Bounded} {Treewidth}},
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isbn = {978-3-642-15369-3},
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isbn = {978-3-642-15369-3},
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doi = {10.1007/978-3-642-15369-3_10},
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doi = {10.1007/978-3-642-15369-3_10},
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language = {en},
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language = {en},
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booktitle = {Approximation, {Randomization}, and {Combinatorial} {Optimization}. {Algorithms} and {Techniques}},
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booktitle = {Approximation, {Randomization}, and {Combinatorial} {Optimization}. {Algorithms} and {Techniques}},
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publisher = {Springer},
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publisher = {Springer},
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author = {Chlamtac, Eden and Krauthgamer, Robert and Raghavendra, Prasad},
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author = {Chlamtac, Eden and Krauthgamer, Robert and Raghavendra, Prasad},
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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} }
|
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|
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