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Yu Cong
2025-05-10 15:37:13 +08:00
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@@ -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}