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main.tex
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main.tex
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\DeclareMathOperator*{\opt}{OPT}
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\DeclareMathOperator*{\len}{len}
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\def\scut{\textsc{Sparsest Cut}}
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\def\nonuscut{\textsc{Non-Uniform Sparsest Cut}}
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\newcommand{\scut}{\textsc{Sparsest Cut}}
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\newcommand{\nonuscut}{\textsc{Non-Uniform Sparsest Cut}}
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\newcommand{\expansion}{\textsc{Expansion}}
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\begin{document}
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\maketitle
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% The present study (the research topic, the research purpose )
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% The information should be presented in coherent paragraphs.
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\scut{} is a fundamental problem in graph algorithms with connections to various cut related problems.
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\begin{problem}[\nonuscut]
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The input is a graph $G=(V,E)$ with edge capacities $c:E\to \R_+$ and a set of vertex pairs $\{s_1,t_1\},\dots,\{s_k,t_k\}$ along with demand values $D_1,\dots,D_k\in \R_+$. The goal is to find a cut $\delta(S)$ of $G$ such that $\frac{c(\delta(S))}{\sum_{i:|S\cap \set{s_i,t_i}|=1}D_i}$ is minimized.
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\end{problem}
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In other words, \nonuscut{} finds the cut that minimizes its capacity divided by the sum of demands of the vertex pairs it separates. There are two important varients of \nonuscut{}. Note that we always consider unordered pair $\{s_i,t_i\}$, i.e., we do not distinguish $\{s_i,t_i\}$ and $\{t_i,s_i\}$.
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\scut{} is the uniform version of \nonuscut{}. The demand is 1 for every possible vertex pair $\{s_i,t_i\}$. In this case, we can remove from the input the pairs and demands. The goal becomes to minimize $\frac{c(\delta(S))}{|S||V\setminus S|}$.
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\expansion{} further simplifies the objective of \scut{} to $\min_{|S|\leq n/2}\frac{c(\delta(S))}{|S|}$.
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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.
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\subsection{related works}
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\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}
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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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% Data collection: e.g. During the program and first 6 months after the program (Aug. 2023- May. 2024)
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% Data analysis: June 2024- Sept. 2024
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\scut{} and \nonuscut{} \citep{chalermsook_approximating_2024}
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\bibliographystyle{plainnat}
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\bibliography{ref}
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\end{document}
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84
ref.bib
84
ref.bib
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pages = {1--20},
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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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title = {Introduction to Sparsest Cut},
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howpublished = {Lecture notes, UIUC CS 598CSC: Topics in Graph Algorithms},
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year = {2024},
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note = {Accessed on May 9, 2025},
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url = {https://courses.grainger.illinois.edu/cs598csc/fa2024/Notes/lec-sparsest-cut.pdf}
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}
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@article{hoory_expander_2006,
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title = {Expander graphs and their applications},
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volume = {43},
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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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doi = {10.1090/S0273-0979-06-01126-8},
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language = {en},
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number = {04},
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urldate = {2025-05-09},
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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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month = aug,
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year = {2006},
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pages = {439--562},
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}
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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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volume = {39},
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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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doi = {10.1137/080731049},
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language = {en},
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number = {5},
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urldate = {2025-05-09},
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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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month = jan,
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year = {2010},
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pages = {1748--1771},
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}
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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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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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}
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@inproceedings{arora_expander_2004,
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address = {New York, NY, USA},
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series = {{STOC} '04},
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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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url = {https://doi.org/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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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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publisher = {Association for Computing Machinery},
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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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file = {Submitted Version:/Users/congyu/Zotero/storage/GALRKD2A/Arora et al. - 2004 - Expander flows, geometric embeddings and graph partitioning.pdf:application/pdf},
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}
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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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volume = {46},
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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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doi = {10.1145/331524.331526},
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number = {6},
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urldate = {2025-05-09},
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journal = {J. ACM},
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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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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},
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}
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