162 lines
14 KiB
TeX
162 lines
14 KiB
TeX
\documentclass[11pt]{article}
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\usepackage{chao}
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% \usepackage{natbib}
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\title{Sparsest Cut}
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\author{}
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\date{}
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\DeclareMathOperator*{\opt}{OPT}
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\DeclareMathOperator*{\len}{len}
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\newcommand{\scut}{\textsc{Sparsest Cut}}
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\newcommand{\nonuscut}{\textsc{Non-Uniform Sparsest Cut}}
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\newcommand{\expansion}{\textsc{Expansion}}
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\begin{document}
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\maketitle
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% \paragraph{Research problem} What is the best possible approximation rate of linear programming based approximation algorithms for \scut? What about algorithms for planar graphs?
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\section{Introduction}
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% Requirement: The introduction may have the following parts:
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% Establishing the importance (Background),
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% literature review (previous research contributions)
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% Gap/the specific problem
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% The present study (the research topic, the research purpose )
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% The information should be presented in coherent paragraphs.
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\scut{} is a fundamental problem in graph algorithms with connections to various cut related problems.
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\begin{problem}[\nonuscut]
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The input is a graph $G=(V,E)$ with edge capacities $c:E\to \R_+$ and a set of vertex pairs $\{s_1,t_1\},\dots,\{s_k,t_k\}$ along with demand values $D_1,\dots,D_k\in \R_+$. The goal is to find a cut $\delta(S)$ of $G$ such that $\frac{c(\delta(S))}{\sum_{i:|S\cap \set{s_i,t_i}|=1}D_i}$ is minimized.
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\end{problem}
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In other words, \nonuscut{} finds the cut that minimizes its capacity divided by the sum of demands of the vertex pairs it separates. There are two important varients of \nonuscut{}. Note that we always consider unordered pair $\{s_i,t_i\}$, i.e., we do not distinguish $\{s_i,t_i\}$ and $\{t_i,s_i\}$.
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\scut{} is the uniform version of \nonuscut{}. The demand is 1 for every possible vertex pair $\{s_i,t_i\}$. In this case, we can remove from the input the pairs and demands. The goal becomes to minimize $\frac{c(\delta(S))}{|S||V\setminus S|}$.
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\expansion{} further simplifies the objective of \scut{} to $\min_{|S|\leq n/2}\frac{c(\delta(S))}{|S|}$.
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% \subsection{importance and connections}
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These problems are interesting since they are related to central concepts in graph theory and help to design algorithms for hard problems on graph. One connections is expander graphs. The importance of expander graphs is thoroughly surveyed in \cite{hoory_expander_2006}. The optimum of \expansion{} is also known as Cheeger constant or conductance of a graph. \scut{} provides a 2-approximation of Cheeger constant, which is especially important in the context of expander graphs as it is a way to measure the edge expansion of a graph. \nonuscut{} is related to other cut problems such as Multicut and Balanced Separator.
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From a more mathematical perspective, the techniques developed for approximating \scut{} are deeply related to metric embedding, which is another fundamental problem in geometry.
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Besides theoretical interests, \scut{} is useful in practical scenarios such as in image segmentation and in some machine leaning algorithms.
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\subsection{related works}
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\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}.
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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}.
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For an overview of the LP methods for \scut{}, see \url{https://courses.grainger.illinois.edu/cs598csc/fa2024/Notes/lec-sparsest-cut.pdf}.
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% \subsection{open problems}
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% 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 $2^{O(k)}\poly(n)$.
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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.
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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.
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\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{}.
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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
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a 2-approximation algorithm for sparsest cut in treewidth $k$ graph with running time $2^{2^{O(k)}}\poly(n)$.
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\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.
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\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}.
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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}}.
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\section{Approximations}
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Techniques for approximating uniform \scut{} and \nonuscut{}.
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\subsection{LP $\Theta(\log n)$}
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\begin{minipage}{0.47\linewidth}
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\begin{equation}\label{IP}
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\begin{aligned}
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\min& & \frac{\sum_e c_e x_e}{\sum_{i} D_i y_i}& & &\\
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s.t.& & \sum_{e\in p} x_e&\geq y_i & &\forall p\in \mathcal{P}_{s_i,t_i}, \forall i\\
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& & x_e,y_i&\in \{0,1\}
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\end{aligned}
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\end{equation}
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\end{minipage}
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\begin{minipage}{0.47\linewidth}
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\begin{equation}\label{LP}
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\begin{aligned}
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\min& & \sum_e c_e x_e& & &\\
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s.t.& & \sum_i D_iy_i&=1 & &\\
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& & \sum_{e\in p} x_e&\geq y_i & &\forall p\in \mathcal{P}_{s_i,t_i}, \forall i\\
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& & x_e,y_i&>0
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\end{aligned}
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\end{equation}
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\end{minipage}
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\bigskip
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\begin{minipage}{0.47\linewidth}
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\begin{equation}\label{dual}
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\begin{aligned}
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\max& & \lambda& & &\\
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s.t.& & \sum_{p\in\mathcal{P}_{s_i,t_i}} y_p&\geq \lambda D_i & &\forall i\\
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& & \sum_i \sum_{p\in \mathcal{P}_{s_i,t_i}, p\ni e}y_p&\geq c_e & &\forall e\\
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& & y_p&\geq 0
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\end{aligned}
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\end{equation}
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\end{minipage}
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\begin{minipage}{0.47\linewidth}
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\begin{equation}\label{metric}
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\begin{aligned}
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\min& & \sum_{uv\in E} c_{uv}d(u,v)& & &\\
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s.t.& & \sum_i D_i d(s_i,t_i)&=1 & &\\
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& & \text{$d$ is a metric on $V$}
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\end{aligned}
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\end{equation}
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\end{minipage}
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\newcommand{\lp}{\texttt{LP\ref{LP}}}
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\newcommand{\ip}{\texttt{IP\ref{IP}}}
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\newcommand{\dual}{\texttt{LP\ref{dual}}}
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\newcommand{\metric}{\texttt{LP\ref{metric}}}
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\begin{enumerate}
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\item \ip{} $\geq$ \lp{}. Given any feasible solution to \ip{}, we can scale all $x_e$ and $y_i$ simultaneously with factor $1/\sum_i D_i y_i$. The scaled solution is feasible for \lp{} and gets the same objective value.
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\item \lp{} $=$ \dual{}. by duality.
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\item \metric{} $=$ \lp{}. It is easy to see \metric{} $\geq$ \lp{} since any feasible metric to \metric{} induces a feasible solution to \lp{}. In fact, the optimal solution to \lp{} also induces a feasible metric. Consider a solution $x_e,y_i$ to \lp{}. Let $d_x$ be the shortest path metric on $V$ using edge length $x_e$. It suffices to show that $y_i=d_x(s_i,t_i)$. This can be seen from a reformulation of \lp{}. The constraint $\sum_i D_i y_i=1$ can be removed and the objective becomes $\sum_e c_e x_e / \sum_i D_i y_i$. This reformulation does not change the optimal solution. Now suppose in the optimal solution to \lp{} there is some $y_i$ which is strictly smaller than $d_x(s_i,t_i)$. Then the denominator $\sum_i D_i y_i$ in the objective of our reformulation can be larger, contradicting to the optimality of solution $x_e,y_i$.
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\end{enumerate}
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\begin{theorem}[Japanese Theorem]
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$D$ is a demand matrix. $D$ is routable in $G$ iff $\forall l:E\to \R_+$, $\sum_e c_e l(e)\geq \sum_{uv} D(u,v) d_l(u,v)$, where $d_l(s,t)$ is the short path distance induced by $l(e)$.
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\end{theorem}
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Note that $D$ is routable iff the optimum of the LPs is at least 1. Then the theorem follows directly from \metric{}.
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\paragraph{$\Theta(\log n)$ flow-cut gap}
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The flow-cut gap is $\opt(\ip{})/\opt(\lp{})$ \cite{leighton_multicommodity_1999}.
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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 $1/\opt(\lp{})$.
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For 1 and 2-commodity flow problem the gap is 1 \cite{Ford_Fulkerson_1956,Hu_1963}.
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However, for $k\geq 3$ the gap becomes larger\footnote{\url{https://en.wikipedia.org/wiki/Approximate_max-flow_min-cut_theorem}}.
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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.
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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 $\Omega(\log n)$.
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For the upperbound it suffices to show there exists a cut of ratio $O(f\log n)$.
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\cite{leighton_multicommodity_1999} gave an algorithmic proof based on \metric{}. This can also be proven using metric embedding results.
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We can solve \metric{} in polynomial time and get a metric on $V$. Then there is an embedding of $V$ into $\R^d$ with $\ell_1$ metric such that the distortion is $O(\log n)$.
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Since $\ell_1$ metric is in the cut cone, our metric on $\R^d$ is a conic combination of cut metrics, which implies\footnote{This requires some work. See \url{https://courses.grainger.illinois.edu/cs598csc/fa2024/Notes/lec-sparsest-cut.pdf}} that there is a cut in the conic combination with value at most $O(\log n)\opt(\metric{})$.
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To find such a cut it suffices to compute a conic combination of cut metrics which is exactly our $\ell_1$ metric in $\R^d$. One way to do this is test $(n-1)d$ cuts by observing the followings,
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\begin{enumerate}
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\item Every coordinate of $\R^d$ corresponds to a line metric;
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\item $\ell_1$ metric in $\R^d$ is the sum of those line metrics;
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\item Every line metric on $n$ points can be represented as some conic combination of $n-1$ cut metrics.
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\end{enumerate}
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\begin{remark}
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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{}?
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\end{remark}
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\subsection{SDP $O(\sqrt{\log n})$}
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SDP approximation follows from metric embedding results.
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\bibliographystyle{alpha}
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\bibliography{ref}
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\end{document}
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