diff --git a/main.pdf b/main.pdf index fde243a..3875efa 100644 Binary files a/main.pdf and b/main.pdf differ diff --git a/main.tex b/main.tex index 143e447..28b0052 100644 --- a/main.tex +++ b/main.tex @@ -45,7 +45,7 @@ From a more mathematical perspective, the techniques developed for approximating Besides theoretical interests, \scut{} is useful in practical scenarios such as in image segmentation and in some machine leaning algorithms. \subsection{related works} -\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}. +\nonuscut{} is APX-hard \citep{juliaJACMapxhard} and, assuming the Unique Game Conjecture, has no polynomial time constant factor aproximation algorithm\citep{chawla_hardness_2005}. \scut{} admits no PTAS \citep{uniformhardnessFocs07}, assuming a widely believed conjecture. 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 treewidth \citep{chlamtac_approximating_2010,gupta2013sparsestcutboundedtreewidth, Chalermsook_2024}. For an overview of the LP methods for \scut{}, see \citep{sparsest_cut_notes}. @@ -61,6 +61,13 @@ The seminal work of \cite{leighton_multicommodity_1999} starts this line of rese 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{}. + +There is also plenty of research concerning \scut{} on some graph classes, for example \citep{bonsma_complexity_2012}. One of the most popular class is graphs with constant treewidth. \citep{Chalermsook_2024} gave a $O(k^2)$ approximation algorithm with complexity $2^{O(k)}\poly(n)$. \citep{Cohen-Addad_Mömke_Verdugo_2024} obtained +a 2-approximation algorithm for sparsest cut in treewidth $k$ graph with running time $2^{2^{O(k)}}\poly(n)$. + +\scut{} is easy on trees and the flow-cut gap is 1 for trees. One explaination mentioned in \citep{sparsest_cut_notes} is that shortest path distance in trees is an $\ell_1$ metric. There are works concerning planar graphs and more generally graphs with constant genus. +\citep{leighton_multicommodity_1999} provided a $\Omega(\log n)$ lowerbound for flow-cut gap for \scut{}. However, it is conjectured that the gap is $O(1)$, while currently the best upperbound is still $O(\sqrt{\log n})$ \citep{rao_small_1999}. +For graphs with constant genus, \citep{lee_genus_2010} gives a $O(\sqrt{\log g})$ approximation for \scut{}, where $g$ is the genus of the input graph. For flow-cut gap in planar graphs the techniques are mainly related to metric embedding theory. \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: @@ -71,12 +78,16 @@ For \nonuscut{} \citep{leighton_multicommodity_1999} only guarantees a $O(\log^2 % Data analysis (the specific data analysis method) % e.g. Using SPSS to analyze the survey data and Using NVivo to analyze the interview data (details of the method and reasons for the choice) % The significance/ implications of the study - +\paragraph{Research type:} theoretical research +\paragraph{Possible difficulties:} The technical depth of open problems in \scut{} might be larger than I expected. If I have no idea after thoroughly understanding metric embedding methods and SDP relaxation, I will immediately move to other problems. \section{Time Table} % Data collection: e.g. During the program and first 6 months after the program (Aug. 2023- May. 2024) % Data analysis: June 2024- Sept. 2024 +understanding existing methods: 2 weeks.\newline +solving a problem or imporving some approximation: at most 2 months. + \bibliographystyle{plainnat} \bibliography{ref} \end{document} diff --git a/ref.bib b/ref.bib index cfdc7a6..e70af77 100644 --- a/ref.bib +++ b/ref.bib @@ -1,21 +1,4 @@ -@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} -} @misc{sparsest_cut_notes, author = {Chekuri, Chandra}, title = {Introduction to Sparsest Cut}, @@ -42,7 +25,7 @@ } @article{arora_osqrtlogn_2010, - title = {\${O}({\textbackslash}sqrt\{{\textbackslash}logn\})\$ {Approximation} to {SPARSEST} {CUT} in \${\textbackslash}tilde\{{O}\}(n{\textasciicircum}2)\$ {Time}}, + title = { {$O(\sqrt{\log n})$} {Approximation} to {SPARSEST} {CUT} in {$\tilde{O}(n^2)$} {Time}}, volume = {39}, issn = {0097-5397, 1095-7111}, url = {http://epubs.siam.org/doi/10.1137/080731049}, @@ -187,7 +170,7 @@ } @article{lognGapAumann98, author = {Aumann, Yonatan and Rabani, Yuval}, - title = {An O(log k) Approximate Min-Cut Max-Flow Theorem and Approximation Algorithm}, + title = {An {$O(\log k)$} Approximate Min-Cut Max-Flow Theorem and Approximation Algorithm}, journal = {SIAM Journal on Computing}, volume = {27}, number = {1}, @@ -198,3 +181,47 @@ 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} } + +@article{Cohen-Addad_Mömke_Verdugo_2024, title={A 2-approximation for the bounded treewidth sparsest cut problem in $\textsf{FPT}$Time}, volume={206}, ISSN={1436-4646}, url={https://doi.org/10.1007/s10107-023-02044-1}, DOI={10.1007/s10107-023-02044-1}, number={1}, journal={Mathematical Programming}, author={Cohen-Addad, Vincent and Mömke, Tobias and Verdugo, Victor}, year={2024}, month=jul, pages={479–495}, language={en} } +@INPROCEEDINGS{uniformhardnessFocs07, + author={Ambuhl, Christoph and Mastrolilli, Monaldo and Svensson, Ola}, + booktitle={48th Annual IEEE Symposium on Foundations of Computer Science (FOCS'07)}, + title={Inapproximability Results for Sparsest Cut, Optimal Linear Arrangement, and Precedence Constrained Scheduling}, + year={2007}, + volume={}, + number={}, + pages={329-337}, + keywords={Cost function;Processor scheduling;Polynomials;NP-complete problem;Approximation algorithms;Single machine scheduling;Computer science}, + doi={10.1109/FOCS.2007.40}} + +@article{bonsma_complexity_2012, + title = {The complexity of finding uniform sparsest cuts in various graph classes}, + volume = {14}, + copyright = {https://www.elsevier.com/tdm/userlicense/1.0/}, + issn = {15708667}, + url = {https://linkinghub.elsevier.com/retrieve/pii/S1570866711001110}, + doi = {10.1016/j.jda.2011.12.008}, + language = {en}, + urldate = {2025-05-10}, + journal = {Journal of Discrete Algorithms}, + author = {Bonsma, Paul and Broersma, Hajo and Patel, Viresh and Pyatkin, Artem}, + month = jul, + year = {2012}, + pages = {136--149}, +} + +@inproceedings{rao_small_1999, + address = {Miami Beach Florida USA}, + title = {Small distortion and volume preserving embeddings for planar and {Euclidean} metrics}, + isbn = {978-1-58113-068-3}, + url = {https://dl.acm.org/doi/10.1145/304893.304983}, + doi = {10.1145/304893.304983}, + language = {en}, + urldate = {2025-05-10}, + booktitle = {Proceedings of the fifteenth annual symposium on {Computational} geometry}, + publisher = {ACM}, + author = {Rao, Satish}, + month = jun, + year = {1999}, + pages = {300--306}, +}