flow-cut gap is Theta(log n)

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@@ -65,7 +65,7 @@ a 2-approximation algorithm for sparsest cut in treewidth $k$ graph with running
 
 \scut{} is easy on trees and the flow-cut gap is 1 for trees. One explaination mentioned in \cite{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.
 \cite{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 $O(\sqrt{\log n})$ \cite{rao_small_1999}.
-For graphs with constant genus, \cite{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.
+For graphs with constant genus, \cite{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\footnote{\url{https://home.ttic.edu/~harry/teaching/teaching.html}}.
 
 
 \section{LP}
@@ -135,7 +135,11 @@ For 1 and 2-commodity flow problem the gap is 1 \cite{Ford_Fulkerson_1956,Hu_196
 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{}.
+\paragraph{$\Theta(\log n)$ flow-cut gap} 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 $f\leq \frac{3n}{\binom{n}{2}(\log n -2)}$ and the gap is therefore $\Omega(\log n)$.
+For the upperbound it suffices to show there exists a cut of ratio $O(f\log n)$.
+\cite{leighton_multicommodity_1999} gave an algorithmic proof based on \metric{}. This can also be proven using metric embedding theorem \cite{sparsest_cut_notes}. (I believe the later method is more general and works for \nonuscut{}, while the former method is limited to uniform \scut{}. However, the proof in \cite{leighton_multicommodity_1999} may have connections with the proof of Bourgain's thm?)
+
+\paragraph{$O(\sqrt{\log n})$ approximation}
 
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