diff --git a/main.pdf b/main.pdf index 46723ec..b38b03b 100644 Binary files a/main.pdf and b/main.pdf differ diff --git a/main.tex b/main.tex index 500bd9a..f1b6bc0 100644 --- a/main.tex +++ b/main.tex @@ -1,4 +1,4 @@ -\documentclass[11pt]{article} +\documentclass[12pt]{article} \usepackage{chao} \usepackage{natbib} @@ -41,7 +41,10 @@ In other words, \nonuscut{} finds the cut that minimizes its capacity divided by These problems are interesting since they are related to central concepts in graph theory and help to design algorithms for hard problems on graph. One connections is expander graphs. The importance of expander graphs is thoroughly surveyed in \citep{hoory_expander_2006}. The optimum of \expansion{} is also known as Cheeger constant or conductance of a graph. \scut{} provides a 2-approximation of Cheeger constant, which is especially important in the context of expander graphs as it is a way to measure the edge expansion of a graph. \nonuscut{} is related to other cut problems such as Multicut and Balanced Separator. \subsection{related works} -\scut{} is generally hard. The currently best approximation algorithm has ratio $O(\sqrt{\log n})$ and running time $\tilde{O}(n^2)$ \citep{arora_osqrtlogn_2010}. Prior to this currently optimal result, there is a long line of research optimizing both the approximation ratio and the complexity, see \citep{arora_expander_2004,leighton_multicommodity_1999} +\scut{} is APX-hard \citep{juliaJACMapxhard} and, assuming the Unique Game Conjecture, has no polynomial time constant factor aproximation algorithm\citep{chawla_hardness_2005}. The currently best approximation algorithm has ratio $O(\sqrt{\log n})$ and running time $\tilde{O}(n^2)$ \citep{arora_osqrtlogn_2010}. Prior to this currently optimal result, there is a long line of research optimizing both the approximation ratio and the complexity, see \citep{arora_expander_2004,leighton_multicommodity_1999}. +There are also works concerning approximating \scut{} on special graph classes such as planar graphs \citep{lee_genus_2010}, graphs with low tree width \citep{gupta2013sparsestcutboundedtreewidth, Chalermsook_2024}. + +One major open problem for \scut{} is the best approximation ratio for planar graphs. It is conjectured that the ratio for planar graphs is $O(1)$ but currently the best lowerbound is $O(\sqrt{\log n})$. \section{Literature Review} % Requirement: summarize previous research contributions and identify the gap or the specific problem diff --git a/ref.bib b/ref.bib index 478137a..626f4df 100644 --- a/ref.bib +++ b/ref.bib @@ -82,7 +82,6 @@ month = jun, year = {2004}, pages = {222--231}, - file = {Submitted Version:/Users/congyu/Zotero/storage/GALRKD2A/Arora et al. - 2004 - Expander flows, geometric embeddings and graph partitioning.pdf:application/pdf}, } @article{leighton_multicommodity_1999, @@ -98,5 +97,74 @@ month = nov, year = {1999}, pages = {787--832}, - file = {Full Text PDF:/Users/congyu/Zotero/storage/EVDSDZAH/Leighton and Rao - 1999 - Multicommodity max-flow min-cut theorems and their use in designing approximation algorithms.pdf:application/pdf}, +} + +@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}, +} +@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}, +} +@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} } +@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} +} + +@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}, }