the Omega(log n) lb

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}

ref.bib

@@ -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}
+}