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z1)=Pc3zN=82Bn3LEB6nT)pFySTzo<$H+>>j)H3!*7S^^;y<6$ryJzt{drm)R#e~u( zCrF|s?dl3!IA35%{S@MHm5{>KBYALyDg~w14{s+apk4t_*N0Q?S6zW*lZRW+MK}ZI z0&sF+m}6Dq)>5mnHD7YQYa$GBPeCq%`8lEA^${0=2C%|5rMIt~rH1lG`D@=aD7$0{ zQC$u3b?&^NB=2A46{AO&-%wxCyL(J3>~R88c44)Hc;w|zU(u@N1EpGqXm7KpdPJhp z=JV`aRUo&9IZRa4!buIO#J?lIus>PeOdqCCvS$VIf&V2eS*6VqXK<8(ch6DNK`O_yZTBy&^)E{rWvca3YNMD0#t<#8!6o>00KBh7A8hmGBPnaaoGO_ Dd8J4) diff --git a/main.tex b/main.tex index aac07d3..ab32d4b 100644 --- a/main.tex +++ b/main.tex @@ -54,9 +54,9 @@ For an overview of the LP methods for \scut{}, see \cite{sparsest_cut_notes}. % 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 $O(\sqrt{\log n})$. For graphs treewidth $k$, an open problem is that whether there is a 2 approximation algorithm that runs in $2^{O(k)}\poly(n)$. -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 better ratio is still possible through other methods. This paper showed that $O(\log^2 n)$ is the best approximation we can achieve using flow-cut gap. +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 (in fact the tight $\Theta(\log n)$ gap was proven by \cite{aumann_rabani_1995} and \cite{Linial_London_Rabinovich_1995}). 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 better ratio is still possible through other methods. This paper showed that $O(\log^2 n)$ is the best approximation we can achieve using flow-cut gap. -For \nonuscut{} \cite{leighton_multicommodity_1999} only guarantees a $O(\log^2 n)$ approximation. This is further improved by \cite{Linial_London_Rabinovich_1995} and \cite{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. \cite{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. +For \nonuscut{} \cite{leighton_multicommodity_1999} only guarantees a $O(\log^2 n)$ approximation. This is further improved by \cite{Linial_London_Rabinovich_1995} and \cite{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. \cite{lognGapAumann98}, \cite{aumann_rabani_1995} and \cite{Linial_London_Rabinovich_1995} 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. \cite{arora_expander_2004} and \cite{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{}. @@ -127,13 +127,15 @@ $D$ is a demand matrix. $D$ is routable in $G$ iff $\forall l:E\to \R_+$, $\sum_ Note that $D$ is routable iff the optimum of the LPs is at least 1. Then the theorem follows directly from \metric{}. \subsection{Flow-cut gap} -The flow-cut gap is $\opt(\ip{})/\opt(\lp{})$. +The flow-cut gap is $\opt(\ip{})/\opt(\lp{})$ \cite{leighton_multicommodity_1999}. Suppose that $G$ satisfies the cut condition, that is, $c(\delta(S))$ is at least the demand separated by $\delta(S)$ for all $S\subset V$. This implies $\opt(\ip{})\geq 1$ and in this case the largest integrality gap is $1/\opt(\lp{})$. -Note that the cut condition is weaker than the routable condition and it is always satisfiable via scaling numbers. For 1 and 2-commodity flow problem the gap is 1 \cite{Ford_Fulkerson_1956,Hu_1963}. However, for $k\geq 3$ the gap becomes larger\footnote{\url{https://en.wikipedia.org/wiki/Approximate_max-flow_min-cut_theorem}}. +It is mentioned in \cite{leighton_multicommodity_1999} that \cite{schrijver_homotopic_1990} proved if the demand graph does not contain either three disjoint edges or a triangle and a disjoint edge, then the gap is 1. + +\paragraph{$\Theta(\log n)$ gap for uniform \scut{}} For the $\Omega(\log n)$ lowerbound consider an uniform \scut{} instance on some 3-regular graph $G$ with unit capacity. In \cite{leighton_multicommodity_1999} they further required that for any $S\subset V$ and small constant $c$, $|\delta(S)|\geq c \min(|S|,|\bar S|)$. Then the value of the sparsest cut is at least $\frac{c}{n-1}$. Observe that for any fixed vertex $v$, there are at most $n/2$ vertices within distance $\log n-3$ of $v$. Thus at least half of the $\binom{n}{2}$ demand pairs are connected with shortest path of length at least $\log n-2$. To sustain a flow $f$ we need at least $\frac{1}{2}\binom{n}{2}(\log n -2)f\leq 3n/2$. Any feasible flow satisfies $f\leq \frac{3n}{\binom{n}{2}(\log n -2)}\leq$ and the gap is $\Omega(\log n)$. The upperbound part only works for uniform \scut{}. \bibliographystyle{alpha} \bibliography{ref} diff --git a/ref.bib b/ref.bib index eb749fb..fb8643d 100644 --- a/ref.bib +++ b/ref.bib @@ -185,8 +185,8 @@ @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}, + 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={}, @@ -229,3 +229,25 @@ @article{Ford_Fulkerson_1956, title={Maximal Flow Through a Network}, volume={8}, rights={https://www.cambridge.org/core/terms}, ISSN={0008-414X, 1496-4279}, url={https://www.cambridge.org/core/product/identifier/S0008414X00036890/type/journal_article}, DOI={10.4153/CJM-1956-045-5}, journal={Canadian Journal of Mathematics}, author={Ford, L. R. and Fulkerson, D. R.}, year={1956}, pages={399–404}, language={en} } @article{Hu_1963, title={Multi-Commodity Network Flows}, volume={11}, ISSN={0030-364X, 1526-5463}, url={https://pubsonline.informs.org/doi/10.1287/opre.11.3.344}, DOI={10.1287/opre.11.3.344}, number={3}, journal={Operations Research}, author={Hu, T. C.}, year={1963}, month=jun, pages={344–360}, language={en} } + +@article{schrijver_homotopic_1990, + title = {Homotopic {Routing} {Methods}}, + language = {en}, + journal = {Paths, Flows, and VLSI-Layout}, + author = {Schrijver, Alexander}, + year = {1990}, + pages = {329 --371}, +} +@inproceedings{aumann_rabani_1995, +author = {Aumann, Yonatan and Rabani, Yuval}, +title = {Improved bounds for all optical routing}, +year = {1995}, +isbn = {0898713498}, +publisher = {Society for Industrial and Applied Mathematics}, +address = {USA}, +booktitle = {Proceedings of the Sixth Annual ACM-SIAM Symposium on Discrete Algorithms}, +pages = {567–576}, +numpages = {10}, +location = {San Francisco, California, USA}, +series = {SODA '95} +}