diff --git a/main.pdf b/main.pdf index aed7964..fde243a 100644 Binary files a/main.pdf and b/main.pdf differ 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} }