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zO9rQF^YTL5e;R*%5@;=NJ!i*7uhUKnCl+`oKt0Ku#@o&iv(r#=oURYA!QN{ZBbzi% zTYVVmhu5^s|Ef!5w3BZk9NOGd7T^@2n+g6KeBwrO(1ZB^rL($=Evs|<85epeg1?wn;_uL2(>uDkbA(z~2P7}cVyll%Om?hV3kS6|Iv zv{Fg5X}iq8`@FhD>Svfh!S&@^>tlbp^`V5p$rnj>SA*KDwb^lPv$2&$Pb*H6#5$fH z-d!2fJhvK+6{;h&X2G!%Bqo=B8#hh&se$|FMaOD|6y*p)xhU!mIa~l>A`lf57Q<&} JSI|(z|1VqS>3RSF diff --git a/main.tex b/main.tex index bcb7328..4f0612a 100644 --- a/main.tex +++ b/main.tex @@ -45,10 +45,10 @@ 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} -\nonuscut{} is APX-hard \cite{juliaJACMapxhard} and, assuming the Unique Game Conjecture, has no polynomial time constant factor aproximation algorithm\cite{chawla_hardness_2005}. \scut{} admits no PTAS \cite{uniformhardnessFocs07}, assuming a widely believed conjecture. The currently best approximation algorithm has ratio $O(\sqrt{\log n})$ and running time $\tilde{O}(n^2)$ \cite{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 \cite{arora_expander_2004,leighton_multicommodity_1999}. +\nonuscut{} is APX-hard \cite{juliaJACMapxhard} and, assuming the Unique Game Conjecture, has no polynomial time constant factor aproximation algorithm\cite{chawla_hardness_2005}. \scut{} admits no PTAS \cite{uniformhardnessFocs07}, assuming a widely believed conjecture. The currently best approximation algorithm for \scut{} has ratio $O(\sqrt{\log n})$ and running time $\tilde{O}(n^2)$ \cite{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 \cite{arora_expander_2004,leighton_multicommodity_1999}. There are also works concerning approximating \scut{} on special graph classes such as planar graphs \cite{lee_genus_2010}, graphs with low treewidth \cite{chlamtac_approximating_2010,gupta2013sparsestcutboundedtreewidth, Chalermsook_2024}. -For an overview of the LP methods for \scut{}, see \cite{sparsest_cut_notes}. +For an overview of the LP methods for \scut{}, see \url{https://courses.grainger.illinois.edu/cs598csc/fa2024/Notes/lec-sparsest-cut.pdf}. % \subsection{open problems} @@ -63,12 +63,14 @@ For \nonuscut{} \cite{leighton_multicommodity_1999} only guarantees a $O(\log^2 There is also plenty of research concerning \scut{} on some graph classes, for example \cite{bonsma_complexity_2012}. One of the most popular class is graphs with constant treewidth. \cite{Chalermsook_2024} gave a $O(k^2)$ approximation algorithm with complexity $2^{O(k)}\poly(n)$. \cite{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 \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. +\scut{} is easy on trees and the flow-cut gap is 1 for trees. One explaination\footnote{\url{https://courses.grainger.illinois.edu/cs598csc/fa2024/Notes/lec-sparsest-cut.pdf}} 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\footnote{\url{https://home.ttic.edu/~harry/teaching/teaching.html}}. -\section{LP} +\section{Approximations} +Techniques for approximating uniform \scut{} and \nonuscut{}. +\subsection{LP $\Theta(\log n)$} \begin{minipage}{0.47\linewidth} \begin{equation}\label{IP} @@ -126,7 +128,7 @@ $D$ is a demand matrix. $D$ is routable in $G$ iff $\forall l:E\to \R_+$, $\sum_ \end{theorem} 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} +\paragraph{$\Theta(\log n)$ flow-cut gap} 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 $1/\opt(\lp{})$. @@ -135,11 +137,12 @@ 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{$\Theta(\log n)$ 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 $\Omega(\log n)$ 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?) +\cite{leighton_multicommodity_1999} gave an algorithmic proof based on \metric{}. This can also be proven using metric embedding theorem, see \url{https://courses.grainger.illinois.edu/cs598csc/fa2024/Notes/lec-sparsest-cut.pdf}. (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? why does the method in \cite{leighton_multicommodity_1999} fail to work on \nonuscut{}?) -\paragraph{$O(\sqrt{\log n})$ approximation} +\subsection{SDP $O(\sqrt{\log n})$} +SDP approximation follows from metric embedding results. \bibliographystyle{alpha} \bibliography{ref} diff --git a/ref.bib b/ref.bib index fb8643d..71e2685 100644 --- a/ref.bib +++ b/ref.bib @@ -1,12 +1,4 @@ -@misc{sparsest_cut_notes, - author = {Chekuri, Chandra}, - title = {Introduction to Sparsest Cut}, - howpublished = {Lecture notes, UIUC CS 598CSC: Topics in Graph Algorithms}, - year = {2024}, - note = {Accessed on May 9, 2025}, - url = {https://courses.grainger.illinois.edu/cs598csc/fa2024/Notes/lec-sparsest-cut.pdf} -} @article{hoory_expander_2006, title = {Expander graphs and their applications},