The integrality gap of the Goemans--Linial SDP relaxation for Sparsest Cut is at least a constant multiple of $\sqrt{\log n}$

TL;DR

This paper establishes that the Goemans--Linial SDP relaxation for the Sparsest Cut problem has an integrality gap that grows at least proportionally to the square root of the logarithm of the number of vertices, indicating a fundamental limit of this approach.

Contribution

It proves a lower bound of ( ext{log} n) on the integrality gap of the Goemans--Linial SDP relaxation for the Sparsest Cut problem, advancing understanding of approximation limits.

Findings

Integrality gap is at least ( ext{log} n)

Limits the effectiveness of the SDP relaxation for large graphs

Provides a fundamental bound for approximation algorithms

Abstract

We prove that the integrality gap of the Goemans--Linial semidefinite programming relaxation for the Sparsest Cut Problem is $\Omega(\sqrt{\log n})$ on inputs with $n$ vertices.

Equations

$$\mathsf{OPT}(C,D)\stackrel{\mathrm{def}}{=}\underset{\varnothing ⊊A⊊\{1,\dots ,n\}}{\min }\frac{{\sum }_{(i,j)\in A\times (\{1,\dots ,n\}\setminus A)}{C}_{ij}}{{\sum }_{(i,j)\in A\times (\{1,\dots ,n\}\setminus A)}{D}_{ij}}.$$

$$\mathsf{ALG}(C,D)⩽\mathsf{OPT}(C,D)⩽\text{uprho}(n)\mathsf{ALG}(C,D),$$

$$\mathsf{SDP}(C,D)\stackrel{\mathrm{def}}{=}\underset{(v_1,\dots ,v_n)\in {\mathsf{NEG}}_n}{\inf }\frac{{\sum }_{i=1}^n{\sum }_{j=1}^n{C}_{ij}\Vert v_i-v_j{\Vert }_2^2}{{\sum }_{i=1}^n{\sum }_{j=1}^n{D}_{ij}\Vert v_i-v_j{\Vert }_2^2},$$

$${\mathsf{NEG}}_n$$

$$\Vert v_i-v_j{\Vert }_2^2⩽\Vert v_i-v_k{\Vert }_2^2+\Vert v_k-v_j{\Vert }_2^2$$

$$\displaystyle\qquad\qquad\mathrm{for\ all\ }i,j,k\in\{1,\ldots,n\}\Big{\}}.$$

$${\text{uprho}}_{\mathsf{GL}}(n)\stackrel{\mathrm{def}}{=}\underset{\begin{array}{c}C,D\in M_n([0,\infty ))\\ C,D\mathrm{symmetric}\end{array}}{\sup }\frac{\mathsf{OPT}(C,D)}{\mathsf{SDP}(C,D)}.$$

$${\mathbb{H}}_{\mathbb{Z}}^5\stackrel{\mathrm{def}}{=}\left\{\left(\begin{array}{cccc}1& \mathsf{a}& \mathsf{b}& \mathsf{e}\\ 0& 1& 0& \mathsf{c}\\ 0& 0& 1& \mathsf{d}\\ 0& 0& 0& 1\end{array}\right):\mathsf{a},\mathsf{b},\mathsf{c},\mathsf{d},\mathsf{e}\in \mathbb{Z}\right\}\subseteq {\mathrm{GL}}_4(\mathbb{R}).$$

$$S\stackrel{\mathrm{def}}{=}\{X_1,X_1^{-1},X_2,X_2^{-1},Y_1,Y_1^{-1},Y_2,Y_2^{-1}\},$$

$$\begin{array}{rl}{X}_1& \stackrel{\mathrm{def}}{=}\left(\begin{array}{cccc}1& 1& 0& 0\\ 0& 1& 0& 0\\ 0& 0& 1& 0\\ 0& 0& 0& 1\end{array}\right),X_2\stackrel{\mathrm{def}}{=}\left(\begin{array}{cccc}1& 0& 1& 0\\ 0& 1& 0& 0\\ 0& 0& 1& 0\\ 0& 0& 0& 1\end{array}\right),\\ Y_1& \stackrel{\mathrm{def}}{=}\left(\begin{array}{cccc}1& 0& 0& 0\\ 0& 1& 0& 1\\ 0& 0& 1& 0\\ 0& 0& 0& 1\end{array}\right),Y_2\stackrel{\mathrm{def}}{=}\left(\begin{array}{cccc}1& 0& 0& 0\\ 0& 1& 0& 0\\ 0& 0& 1& 1\\ 0& 0& 0& 1\end{array}\right).\end{array}$$

$$\begin{gathered} (\mathsf{a},\mathsf{b},\mathsf{c},\mathsf{d},\mathsf{e})(\text{upalpha},\text{upbeta},\text{upgamma},\text{updelta},\text{upepsilon}) \\ \stackrel{\mathrm{def}}{=}(\mathsf{a}+\text{upalpha},\mathsf{b}+\text{upbeta},\mathsf{c}+\text{upgamma},\mathsf{d}+\text{updelta},\mathsf{e}+\text{upepsilon}+\mathsf{a}\text{upgamma}+\mathsf{b}\text{updelta}). \end{gathered}$$

$$\upomega\big{(}d_{W}(x,y)\big{)}\lesssim\|\upphi(x)-\upphi(y)\|_{1}\leqslant d_{W}(x,y),$$

$${\int }_1^{cR}\frac{\text{upomega}(s{)}^2}{s^3}\mathrm{d}s\lesssim 1.$$

$$\forall\,x,y\in\mathbb{Z}^{5},\quad\upomega\big{(}d_{W}(x,y)\big{)}\lesssim\|\upphi(x)-\upphi(y)\|_{1}\leqslant d_{W}(x,y),$$

$${\int }_1^{\infty }\frac{\text{upomega}(s{)}^2}{s^3}\mathrm{d}s\lesssim 1.$$

$$\text{upomega}(t)\lesssim \frac{t}{\sqrt{(\log t)\log \log t}}.$$

$$c_{1}\big{(}\mathcal{B}_{R},d_{W}^{1-\varepsilon}\big{)}\asymp\min\left\{\frac{1}{\sqrt{\varepsilon}},\sqrt{\log R}\right\}.$$

$$\partial_{\mathsf{h}}\Omega\stackrel{{\scriptstyle\mathrm{def}}}{{=}}\big{\{}(x,y)\in\Omega\times\left(\mathbb{Z}^{5}\smallsetminus\Omega\right):x^{-1}y\in S\big{\}}.$$

$$\partial^{t}_{\mathsf{v}}\Omega\stackrel{{\scriptstyle\mathrm{def}}}{{=}}\Big{\{}(x,y)\in\Omega\times\left(\mathbb{Z}^{5}\smallsetminus\Omega\right):x^{-1}y\in\left\{Z^{t},Z^{-t}\right\}\Big{\}}.$$

$$|\partial_{\mathsf{v}}\Omega|\stackrel{{\scriptstyle\mathrm{def}}}{{=}}\bigg{(}\sum_{t=1}^{\infty}\frac{|\partial^{t}_{\mathsf{v}}\Omega|^{2}}{t^{2}}\bigg{)}^{\frac{1}{2}}.$$

$$\left(\sum_{t=1}^{\infty}\frac{1}{t^{2}}\bigg{(}\sum_{h\in\mathbb{Z}^{5}}\big{\|}\upphi\big{(}hZ^{t}\big{)}-\upphi(h)\big{\|}_{1}\bigg{)}^{2}\right)^{\frac{1}{2}}\\ \lesssim\sum_{h\in\mathbb{Z}^{5}}\sum_{\upsigma\in S}\big{\|}\upphi(h\upsigma)-\upphi(h)\big{\|}_{1}.$$

$$\left(\sum_{t=1}^{n^{2}}\frac{1}{t^{2}}\bigg{(}\sum_{h\in\mathcal{B}_{n}}\big{\|}\upphi\big{(}hZ^{t}\big{)}-\upphi(h)\big{\|}_{1}\bigg{)}^{2}\right)^{\frac{1}{2}}\\ \lesssim\sum_{h\in\mathcal{B}_{\upalpha n}}\sum_{\upsigma\in S}\big{\|}\upphi(h\upsigma)-\upphi(h)\big{\|}_{1}.$$

$$\displaystyle\sum_{t=1}^{n^{2}}\frac{1}{t^{2}}\bigg{(}\sum_{h\in\mathcal{B}_{n}}\big{\|}\upphi\big{(}hZ^{t}\big{)}-\upphi(h)\big{\|}_{1}\bigg{)}^{2}\gtrsim\sum_{t=1}^{n^{2}}\frac{|\mathcal{B}_{n}|^{2}\upomega\big{(}\upbeta\sqrt{t}\big{)}^{2}}{t^{2}}$$

$$\displaystyle\gtrsim n^{12}\sum_{t=1}^{n^{2}}\int_{t}^{t+1}\frac{\upomega\big{(}\upbeta\sqrt{u/2}\big{)}^{2}}{u^{2}}\,\mathrm{d}u=\upbeta^{2}n^{12}\int_{\frac{\upbeta}{\sqrt{2}}}^{\frac{\upbeta\sqrt{n^{2}+1}}{\sqrt{2}}}\frac{\upomega(s)^{2}}{s^{3}}\,\mathrm{d}s$$

$$⩾\frac{{\text{upbeta}}^2(R/2{)}^{12}}{\max \{(1+\text{upgamma}{)}^{12},{\text{upalpha}}^{12}\}}{\int }_1^{\frac{\text{upbeta}R}{2\max \{1+\text{upgamma},\text{upalpha}\}}}\frac{\text{upomega}(s{)}^2}{s^3}\mathrm{d}s,$$

$$\begin{gathered} uv\stackrel{\mathrm{def}}{=}u+v \\ +\frac{x_1(u)y_1(v)-y_1(u)x_1(v)+x_2(u)y_2(v)-y_2(u)x_2(v)}{2}Z. \end{gathered}$$

$${\mathfrak{s}}_{\text{uptheta}}({\text{upalpha}}_1,{\text{upalpha}}_2,{\text{upbeta}}_1,{\text{upbeta}}_2,\text{upgamma})=(\text{uptheta}{\text{upalpha}}_1,\text{uptheta}{\text{upalpha}}_2,\text{uptheta}{\text{upbeta}}_1,\text{uptheta}{\text{upbeta}}_2,{\text{uptheta}}^2\text{upgamma})$$

$$d(h,0)⩽|x_1(h)|+|x_2(h)|+|y_1(h)|+|y_2(h)|+4\sqrt{|z(h)|}⩽\frac{C}{2}d(h,0).$$

$${\overline{\mathsf{v}}}_U(E)(s)$$

$$\displaystyle=\frac{1}{2^{s}}\int_{U}\Big{|}1_{E}(u)-1_{E}\big{(}uZ^{-2^{2s}}\big{)}\Big{|}\,\mathrm{d}\mathcal{H}^{6}(u).$$

$$\Gamma_{f}\stackrel{{\scriptstyle\mathrm{def}}}{{=}}\Big{\{}vX_{2}^{f(v)}:\ v\in V\Big{\}}\\ \stackrel{{\scriptstyle\eqref{eq:def heisenberg algebra product}}}{{=}}\Big{\{}\Big{(}\mathsf{a},f(\mathsf{a},{\mathsf{c}},\mathsf{d},\mathsf{e}),{\mathsf{c}},\mathsf{d},\mathsf{e}-\frac{1}{2}\mathsf{d}f(\mathsf{a},{\mathsf{c}},\mathsf{d},\mathsf{e})\Big{)}:\ \mathsf{a},{\mathsf{c}},\mathsf{d},\mathsf{e}\in\mathbb{R}\Big{\}},$$

$$\forall w_1,w_2\in {\Gamma }_f,|x_2(w_1)-x_2(w_2)|⩽\text{uplambda}d(w_1,w_2).$$

$$\forall\,r\in(0,\infty),\qquad\big{\|}\overline{\mathsf{v}}_{B_{r}}\big{(}\Gamma^{+}_{f}\big{)}\big{\|}_{L_{2}(\mathbb{R})}\lesssim\frac{r^{5}}{1-\uplambda}.$$

Full text

The theorem stated above is the culmination of a program that was pursued in the works (Lee and Naor, 2006; Cheeger and Kleiner, 2010a, b; Cheeger et al., 2009, 2011; Austin et al., 2013; Lafforgue and Naor, 2014) whose aim is to understand the performance of the Goemans–Linial semidefinite program through the embeddability properties of Heisenberg groups. These investigations have mathematical significance even beyond their established relevance to approximation algorithms and combinatorial optimization. In particular they contribute to a range of mathematical disciplines including functional analysis, geometric group theory, harmonic analysis, sub-Riemannian geometry, geometric measure theory, ergodic theory, group representations, and metric differentiation. This article builds on the above cited works, with the “twist” that while those works were equally valid for any finite dimensional Heisenberg group, our result holds for the Heisenberg group of dimension $5$ (or higher) but fails for the $3$-dimensional Heisenberg group. This insight leads to our core contribution, which is a deduction of an endpoint $L_{1}$-boundedness of a certain singular integral on $\mathbb{R}^{5}$ from the (local) $L_{2}$-boundedness of the corresponding singular integral on $\mathbb{R}^{3}$. To do this, we devise a corona-type decomposition of subsets of a Heisenberg group, in the spirit of the construction (David and Semmes, 1993) that David and Semmes performed in $\mathbb{R}^{n}$, but with two main conceptual differences (in addition to more technical differences that arise from the peculiarities of the geometry of Heisenberg group). Firstly, the“atoms” of our decomposition are perturbations of intrinsic Lipschitz graphs in the sense of Franchi, Serapioni, and Serra Cassano (Franchi et al., 2006) (plus the requisite “wild” regions that satisfy a Carleson packing condition). Secondly, we control the local overlap of our corona decomposition by using quantitative monotonicity rather than Jones-type $\upbeta$-numbers.

Sparsest Cut Problem, approximation algorithms, semidefinite programming, metric embeddings.

††copyright: rightsretained††doi: 10.1145/3055399.3055413††isbn: 978-1-4503-4528-6/17/06††conference: 49th Annual ACM SIGACT Symposium on the Theory of Computing; June 2017; Montreal, Canada††journalyear: 2017††price: 15.00††ccs: Theory of computation Approximation algorithms analysis††ccs: Theory of computation Mathematical optimization

1. Introduction

Fix $n\in\mathbb{N}$. The input of the Sparsest Cut Problem consists of two $n$ by $n$ symmetric matrices with nonnegative entries $C=(C_{ij}),D=(D_{ij})\in M_{n}([0,\infty))$, which are commonly called capacities and demands, respectively. The goal is to design a polynomial-time algorithm to evaluate the quantity

[TABLE]

In view of the extensive literature on the Sparsest Cut Problem, it would be needlessly repetitive to recount here the rich and multifaceted impact of this optimization problem on computer science and mathematics; see instead the articles (Agrawal et al., 1990; Leighton and Rao, 1999), the surveys (Shmoys, 1997; Linial, 2002a; Chawla, 2008; Naor, 2010), Chapter 10 of the monograph (Deza and Laurent, 1997), Chapter 15 of the monograph (Matoušek, 2002a), Chapter 1 of the monograph (Ostrovskii, 2013), and the references therein. It suffices to say that by tuning the choice of matrices $C,D$ to the problem at hand, the minimization in (1) finds a partition of the “universe” $\{1,\ldots,n\}$ into two parts, namely the sets $A$ and $\{1,\ldots,n\}\smallsetminus A$, whose appropriately weighted interface is as small as possible, thus allowing for inductive solutions of various algorithmic tasks, a procedure known as divide and conquer. (Not all of the uses of the Sparsest Cut Problem fit into this framework. A recent algorithmic application of a different nature can be found in (Makarychev et al., 2014).)

It is $NP$-hard to compute $\mathsf{OPT}(C,D)$ in polynomial time (Shahrokhi and Matula, 1990). By (Chuzhoy and Khanna, 2009) there exists $\varepsilon_{0}>0$ such that it is even $NP$-hard to compute $\mathsf{OPT}(C,D)$ within a multiplicative factor of less than $1+\varepsilon_{0}$. If one assumes Khot’s Unique Games Conjecture (Khot, 2002, 2010; Trevisan, 2012) then by (Chawla et al., 2006; Khot and Vishnoi, 2015) there does not exist a polynomial-time algorithm that can compute $\mathsf{OPT}(C,D)$ within any universal constant factor.

By the above hardness results, a much more realistic goal would be to design a polynomial-time algorithm that takes as input the capacity and demand matrices $C,D\in M_{n}([0,\infty))$ and outputs a number $\mathsf{ALG}(C,D)$ that is guaranteed to satisfy

[TABLE]

with (hopefully) the quantity $\uprho(n)$ growing to $\infty$ slowly as $n\to\infty$. Determining the best possible asymptotic behaviour of $\uprho(n)$ (assuming $P\neq NP$) is an open problem of major importance.

In (Linial et al., 1995; Aumann and Rabani, 1998) an algorithm was designed, based on linear programming (through the connection to multicommodity flows) and Bourgain’s embedding theorem (Bourgain, 1985), which yields $\uprho(n)=O(\log n)$. An algorithm based on semidefinite programming (to be described precisely below) was proposed by Goemans and Linial in the mid-1990s. To the best of our knowledge this idea first appeared in the literature in (Goemans, 1997, page 158), where it was speculated that it might yield a constant factor approximation for Sparsest Cut (see also (Linial, 2002a, b)). In what follows, we denote the approximation ratio of the Goemans–Linial algorithm on inputs of size at most $n$ by $\uprho_{\mathsf{GL}}(n)$. The hope that $\uprho_{\mathsf{GL}}(n)=O(1)$ was dashed in the remarkable work (Khot and Vishnoi, 2015), where the lower bound $\uprho_{\mathsf{GL}}(n)\gtrsim\sqrt[6]{\log\log n}$ was proven.111Here, and in what follows, we use the following (standard) asymptotic notation. Given $a,b>0$, the notations $a\lesssim b$ and $b\gtrsim a$ mean that $a\leqslant\mathsf{K}b$ for some universal constant $\mathsf{K}>0$. The notation $a\asymp b$ stands for $(a\lesssim b)\wedge(b\lesssim a)$. Thus $a\lesssim b$ and $a\gtrsim b$ are the same as $a=O(b)$ and $a=\Omega(b)$, respectively, and $a\asymp b$ is the same as $a=\Theta(b)$. An improved analysis of the ideas of (Khot and Vishnoi, 2015) was conducted in (Krauthgamer and Rabani, 2009), yielding the estimate $\uprho_{\mathsf{GL}}(n)\gtrsim\log\log n$. An entirely different approach based on the geometry of the Heisenberg group was introduced in (Lee and Naor, 2006). In combination with the important works (Cheeger and Kleiner, 2010a, b) it gives a different proof that $\lim_{n\to\infty}\uprho_{\mathsf{GL}}(n)=\infty$. In (Cheeger et al., 2009, 2011) the previously best-known bound $\uprho_{\mathsf{GL}}(n)\gtrsim(\log n)^{\updelta}$ was obtained for an effective (but small) positive universal constant $\updelta$.

Despite these lower bounds, the Goemans–Linial algorithm yields an approximation ratio of $o(\log n)$, so it is asymptotically more accurate than the linear program of (Linial et al., 1995; Aumann and Rabani, 1998). Specifically, in (Chawla et al., 2008) it was shown that $\uprho_{\mathsf{GL}}(n)\lesssim(\log n)^{\frac{3}{4}}$. This was improved in (Arora et al., 2008) to $\uprho_{\mathsf{GL}}(n)\lesssim(\log n)^{\frac{1}{2}+o(1)}$. See Section 1.7 below for additional background on the results quoted above. No other polynomial-time algorithm for the Sparsest Cut problem is known (or conjectured) to have an approximation ratio that is asymptotically better than that of the Goemans–Linial algorithm. However, despite major scrutiny by researchers in approximation algorithms, the asymptotic behavior of $\uprho_{\mathsf{GL}}(n)$ as $n\to\infty$ remained unknown. Theorem 1.1 below resolves this question up to lower-order factors.

Theorem 1.1.

The approximation ratio of the Goemans–Linial algorithm satisfies $\uprho_{\mathsf{GL}}(n)\gtrsim\sqrt{\log n}$.

1.1. The SDP relaxation

The Goemans–Linial algorithm is simple to describe. It takes as input the symmetric matrices $C,D\in M_{n}([0,\infty))$ and proceeds to compute the following quantity.

[TABLE]

where

[TABLE]

Thus $\mathsf{NEG}_{n}$ is the set of $n$-tuples $(v_{1},\ldots v_{n})$ of vectors in $\mathbb{R}^{n}$ such that $(\{v_{1},\ldots,v_{n}\},\upnu_{n})$ is a semi-metric space, where $\upnu_{n}:\mathbb{R}^{n}\times\mathbb{R}^{n}\to[0,\infty)$ is defined by $\upnu_{n}(x,y)=\sum_{j=1}^{n}(x_{j}-y_{j})^{2}=\|x-y\|_{2}^{2}$ for every $x=(x_{1},\ldots,x_{n}),y=(y_{1},\ldots,y_{n})\in\mathbb{R}^{n}$. A semi-metric space $(X,d_{X})$ is said (Deza and Laurent, 1997) to be of negative type if $(X,\sqrt{d_{X}})$ embeds isometrically into a Hilbert space. So, $\mathsf{NEG}_{n}$ can be described as the set of all (ordered) negative type semi-metrics of size $n$. It is simple to check that the evaluation of the quantity $\mathsf{SDP}(C,D)$ can be cast as a semidefinite program (SDP), so it can be achieved (up to $o(1)$ precision) in polynomial time (Grötschel et al., 1993). One has $\mathsf{SDP}(C,D)\leqslant\mathsf{OPT}(C,D)$ for all symmetric matrices $C,D\in M_{n}([0,\infty))$. See e.g. (Matoušek, 2002b, Section 15.9) or (Naor, 2010, Section 4.3) for an explanation of the above assertions about $\mathsf{SDP}(C,D)$, as well as additional background and motivation. The pertinent question is therefore to evaluate the asymptotic behavior as $n\to\infty$ of the sequence

[TABLE]

This is the quantity $\uprho_{\mathsf{GL}}(n)$ appearing in Theorem 1.1, also known as the integrality gap of the Goemans–Linial semidefinite programming relaxation for the Sparsest Cut Problem.

1.2. Bi-Lipschitz embeddings

A duality argument of Rabinovich (see (Naor, 2010, Lemma 4.5) or (Cheeger et al., 2009, Section 1)) establishes that $\uprho_{\mathsf{GL}}(n)$ is equal to the largest possible $L_{1}$-distortion of an $n$-point semi-metric of negative type. If $d:\{1,\ldots,n\}^{2}\to[0,\infty)$ is a semi-metric, its $L_{1}$ distortion, denoted $c_{1}(\{1,\ldots,n\},d)$, is the smallest $D\in[1,\infty)$ for which there are integrable functions222If one wishes to use finite-dimensional vectors rather than functions then by (Witsenhausen, 1986) there exist $v_{1},\ldots,v_{n}\in\mathbb{R}^{n(n-1)/2}$ such that $\int_{0}^{1}|f_{i}(t)-f_{j}(t)|\,\mathrm{d}t=\|v_{i}-v_{j}\|_{1}=\sum_{k=1}^{n(n-1)/2}|v_{ik}-v_{jk}|$ for every $i,j\in\{1,\ldots,n\}$. $f_{1},\ldots,f_{n}:[0,1]\to\mathbb{R}$ such that $\int_{0}^{1}|f_{i}(t)-f_{j}(t)|\,\mathrm{d}t\leqslant d(i,j)\leqslant D\int_{0}^{1}|f_{i}(t)-f_{j}(t)|\,\mathrm{d}t$ for every $i,j\in\{1,\ldots,n\}$. Rabinovich’s duality argument proves that $\uprho_{\mathsf{GL}}(n)$ is equal to the maximum of $c_{1}(\{1,\ldots,n\},d)$ over all possible semi-metrics $d$ of negative type on $\{1,\ldots,n\}$. Hence, Theorem 1.1 is equivalent to the assertion that for every $n\in\mathbb{N}$ there exists a metric of negative type $d:\{1,\ldots,n\}^{2}\to[0,\infty)$ for which $c_{1}(\{1,\ldots,n\},d)\gtrsim\sqrt{\log n}$.

1.3. A poorly-embeddable metric

The $5$-dimensional discrete Heisenberg group, denoted $\mathbb{H}_{\mathbb{Z}}^{5}$, is the following group of $4$ by $4$ invertible matrices, equipped with the usual matrix multiplication.

[TABLE]

This group is generated by the symmetric set

[TABLE]

where

[TABLE]

For notational convenience we shall identify the matrix in (2) with the vector $(\mathsf{a},\mathsf{b},{\mathsf{c}},\mathsf{d},\mathsf{e})\in\mathbb{Z}^{5}$. This yields an identification of $\mathbb{H}_{\mathbb{Z}}^{5}$ with the $5$-dimensional integer grid $\mathbb{Z}^{5}$. We view $\mathbb{Z}^{5}$ as a (noncommutative) group equipped with the product that is inherited from matrix multiplication through the above identification, i.e., for every $(\mathsf{a},\mathsf{b},{\mathsf{c}},\mathsf{d},\mathsf{e}),(\upalpha,\upbeta,\upgamma,\updelta,\upepsilon)\in\mathbb{Z}^{5}$ we set

[TABLE]

Note that under the above identification the identity element of $\mathbb{H}_{\mathbb{Z}}^{5}$ is the zero vector $\mathbf{0}\in\mathbb{Z}^{5}$, the inverse of an element $h=(\mathsf{a},\mathsf{b},{\mathsf{c}},\mathsf{d},\mathsf{e})\in\mathbb{Z}^{5}$ is $h^{-1}=(-\mathsf{a},-\mathsf{b},-{\mathsf{c}},-\mathsf{d},-\mathsf{e}+\mathsf{a}{\mathsf{c}}+\mathsf{b}\mathsf{d})$, and the generators $X_{1},X_{2},Y_{1},Y_{2}$ in (3) are the first four standard basis elements of $\mathbb{R}^{5}$. Let $Z$ denote the fifth standard basis element of $\mathbb{R}^{5}$, i.e., $Z=(0,0,0,0,1)$. We then have the relations $[X_{1},Y_{1}]=[X_{2},Y_{2}]=Z$ and $[X_{1},X_{2}]=[X_{1},Y_{2}]=[X_{1},Z]=[Y_{1},X_{2}]=[Y_{1},Y_{2}]=[Y_{1},Z]=[X_{2},Z]=[Y_{2},Z]=\mathbf{0}$, where we recall the standard commutator notation $[g,h]=ghg^{-1}h^{-1}$ for every two group elements $g,h\in\mathbb{H}_{\mathbb{Z}}^{5}$. In other words, any two elements from $\{X_{1},X_{2},Y_{1},Y_{2},Z\}$ other than $X_{1},Y_{1}$ or $X_{2},Y_{2}$ commute, and the commutators of $X_{1},Y_{1}$ and $X_{2},Y_{2}$ are both equal to $Z$. In particular, $Z$ commutes with all of the members of the generating set $S$, and therefore $Z$ is in the center of $\mathbb{H}_{\mathbb{Z}}^{5}$. It is worthwhile to mention that these commutation relations could be used to define the group $\mathbb{H}_{\mathbb{Z}}^{5}$ abstractly using generators and relations, but this fact will not be needed in what follows.

This group structure induces a graph $\mathcal{X}_{S}(\mathbb{H}_{\mathbb{Z}}^{5})$ on $\mathbb{Z}^{5}$, called the Cayley graph of $\mathbb{H}^{5}_{\mathbb{Z}}$. The edges of this graph are defined to be the unordered pairs of the form $\{h,hs\}$, where $h\in\mathbb{Z}^{5}$ and $s\in S$. This is an $8$-regular connected graph, and by the group law (4), the neighbors of each vertex $(\mathsf{a},\mathsf{b},\mathsf{c},\mathsf{d},\mathsf{e})\in\mathbb{Z}^{5}$ are $(\mathsf{a}\pm 1,\mathsf{b},\mathsf{c},\mathsf{d},\mathsf{e}),(\mathsf{a},\mathsf{b}\pm 1,\mathsf{c},\mathsf{d},\mathsf{e}),(\mathsf{a},\mathsf{b},\mathsf{c}\pm 1,\mathsf{d},\mathsf{e}\pm\mathsf{a}),(\mathsf{a},\mathsf{b},\mathsf{c},\mathsf{d}\pm 1,\mathsf{e}\pm\mathsf{b}).$ The shortest-path metric on $\mathbb{Z}^{5}$ that is induced by this graph structure will be denoted below by $d_{W}:\mathbb{Z}^{5}\times\mathbb{Z}^{5}\to\mathbb{N}\cup\{0\}$. This metric is also known as the left-invariant word metric on the Heisenberg group $\mathbb{H}_{\mathbb{Z}}^{5}$. For every $R\in[0,\infty)$ denote the (closed) ball of radius $R$ centered at the identity element by $\mathcal{B}_{R}=\{h\in\mathbb{Z}^{5}:\ d_{W}(h,\mathbf{0})\leqslant R\}$. It is well-known (see e.g. (Bass, 1972)) that $|\mathcal{B}_{R}|\asymp R^{6}$ and $d_{W}(\mathbf{0},Z^{R})\asymp\sqrt{R}$ for every $R\in\mathbb{N}$. Our main result is the following theorem.

Theorem 1.2.

For all $R\geqslant 2$ we have $c_{1}(\mathcal{B}_{R},d_{W})\asymp\sqrt{\log R}$.

The new content of Theorem 1.2 is the bound $c_{1}(\mathcal{B}_{R},d_{W})\gtrsim\sqrt{\log R}$. The matching upper bound $c_{1}(\mathcal{B}_{R},d_{W})\lesssim\sqrt{\log R}$ has several proofs in the literature; see e.g. the discussion immediately following Corollary 1.3 in (Lafforgue and Naor, 2014) or Section 1.7.2 below. The previous best known estimate (Cheeger et al., 2011) was that there exists a universal constant $\updelta>0$ such that $c_{1}(\mathcal{B}_{R},d_{W})\geqslant(\log R)^{\updelta}$. By (Lee and Naor, 2006, Theorem 2.2) the metric $d_{W}$ is bi-Lipschitz equivalent to a metric on $\mathbb{H}^{5}_{\mathbb{Z}}$ that is of negative type. We remark that (Lee and Naor, 2006) makes this assertion for a different metric on a larger continuous group that contains $\mathbb{H}_{\mathbb{Z}}^{5}$ as a discrete co-compact subgroup, but by a simple general result (e.g. (Burago et al., 2001, Theorem 8.3.19)) the word metric $d_{W}$ is bi-Lipschitz equivalent to the metric considered in (Lee and Naor, 2006). Since $|\mathcal{B}_{R}|\asymp R^{6}$, we have $\sqrt{\log|\mathcal{B}_{R}|}\asymp\sqrt{\log R}$, so Theorem 1.2 implies Theorem 1.1 through the duality result of Rabinovich that was recalled in Section 1.7.2.

The following precise theorem about $L_{1}$ embeddings that need not be bi-Lipschitz implies Theorem 1.2 by considering the special case of the modulus $\upomega(t)=t/D$ for $D\geqslant 1$ and $t\in[0,\infty)$.

Theorem 1.3.

There exists a universal constant $c\in(0,1)$ with the following property. Fix $R\geqslant 2$ and a nondecreasing function $\upomega:[1,\infty)\to[1,\infty)$. Then there exists $\upphi:\mathcal{B}_{R}\to L_{1}$ for which every distinct $x,y\in\mathcal{B}_{R}$ satisfy

[TABLE]

if and only if* $\upomega(t)\lesssim t$ for all $t\in[1,\infty)$ and*

[TABLE]

The fact that the integrability requirement (6) implies the existence of the desired embedding $\upphi$ is due to (Tessera, 2008, Corollary 5). The new content of Theorem 1.3 is that the existence of the embedding $\upphi$ implies (6). By letting $R\to\infty$ in Theorem 1.3 we see that there exists $\upphi:\mathbb{Z}^{5}\to L_{1}$ that satisfies

[TABLE]

if and only if

[TABLE]

In (Cheeger et al., 2011) it was shown that if $\upphi:\mathbb{Z}^{5}\to L_{1}$ satisfies (7), then there must exist arbitrarily large $t\geqslant 2$ for which $\upomega(t)\lesssim t/(\log t)^{\updelta}$, where $\updelta>0$ is a universal constant. This follows from (8) with $\updelta=\frac{1}{2}$, which is the largest possible constant for which this conclusion holds true. This positively answers a question that was asked in (Cheeger et al., 2011, Remark 1.7). In fact, it provides an even better conclusion, because (8) implies that, say, there must exist arbitrarily large $t\geqslant 4$ for which

[TABLE]

(The precise criterion is the integrability condition (8).) Finally, by considering $\upomega(t)=t^{1-\varepsilon}/D$ for $\varepsilon\in(0,1)$ and $D\geqslant 1$, we obtain the following notable corollary.

Corollary 1.4 ($L_{1}$ distortion of snowflakes).

For every $\varepsilon\in(0,1)$ we have c_{1}\big{(}\mathbb{Z}^{5},d_{W}^{1-\varepsilon}\big{)}\asymp\frac{1}{\sqrt{\varepsilon}}.

The fact that for every $O(1)$-doubling metric space $(X,d)$ we have $c_{1}(X,d^{1-\varepsilon})\lesssim 1/\sqrt{\varepsilon}$ follows from an argument of (Lee et al., 2005) (see also (Naor and Silberman, 2011, Theorem 5.2)). Corollary 1.4 shows that this is sharp. More generally, it follows from Theorem 1.3 that for every $R\geqslant 2$ and $\varepsilon\in(0,1)$ we have

[TABLE]

1.4. Vertical-versus-horizontal isoperimetry

Our new non-embeddability results are all consequences of an independently interesting isoperimetric-type inequality which we shall now describe. Roughly speaking, this inequality subtly quantifies the fact that for any $n\in\mathbb{Z}$ and any $h\in\mathbb{H}_{\mathbb{Z}}^{5}$, there are many paths in the Cayley graph $\mathcal{X}_{S}(\mathbb{H}_{\mathbb{Z}}^{5})$ of length roughly $\sqrt{n}$ that connect $h$ to $hZ^{n}$. Consequently, if a finite subset $\Omega\subseteq\mathbb{Z}^{5}$ has a small edge boundary in the Cayley graph, then the number of pairs $(x,y)\in\mathbb{Z}^{5}\times\mathbb{Z}^{5}$ for which $|\{x,y\}\cap\Omega|=1$ yet $x$ and $y$ differ only in their fifth (vertical) coordinate must also be small. It turns out that the proper interpretation of the term “small” is this context is not at all obvious, and it should be measured in a certain multi-scale fashion. Formally, we consider the following quantities.

Definition 1.5 (Discrete boundaries).

For $\Omega\subseteq\mathbb{Z}^{5}$, the horizontal boundary of $\Omega$ is defined by

[TABLE]

Given also $t\in\mathbb{N}$, the $t$-vertical boundary* of $\Omega$ is defined by*

[TABLE]

The horizontal perimeter of $\Omega$ is defined to be the cardinality $|\partial_{\mathsf{h}}\Omega|$ of its horizontal boundary. The vertical perimeter of $\Omega$ is defined to be the quantity

[TABLE]

The horizontal perimeter of $\Omega$ is nothing more than the size of its edge boundary in the Cayley graph $\mathcal{X}_{S}(\mathbb{H}_{\mathbb{Z}}^{5})$. The vertical perimeter of $\Omega$ is a more subtle concept that does not have such a simple combinatorial description. The definition (11) was first published in (Lafforgue and Naor, 2014, Section 4), where the isoperimetric-type conjecture that we resolve here as Theorem 1.6 below also appeared for the first time. These were formulated by the first named author and were circulating for several years before (Lafforgue and Naor, 2014) appeared, intended as a possible route towards the algorithmic application that we indeed succeed to obtain here. That “vertical smallness” should be measured through the quantity $|\partial_{\mathsf{v}}\Omega|$, i.e., the $\ell_{2}$ norm of the sequence $\{|\partial_{\mathsf{v}}^{t}\Omega|/t\}_{t=1}^{\infty}$, was arrived at through trial and error, inspired by functional inequalities that were obtained in (Austin et al., 2013; Lafforgue and Naor, 2014), as explained in (Lafforgue and Naor, 2014, Section 4).

Theorem 1.6.

Every $\Omega\subseteq\mathbb{Z}^{5}$ satisfies $|\partial_{\mathsf{v}}\Omega|\lesssim|\partial_{\mathsf{h}}\Omega|$.

The significance of Theorem 1.6 can only be fully appreciated through an examination of the geometric and analytic reasons for its validity. To facilitate this, we shall include in this extended abstract an extensive overview of the ideas of the proof of Theorem 1.6; see Section 1.6 below. Before doing so, we shall now demonstrate the utility of Theorem 1.6 by using it to deduce Theorem 1.3. As explained above, by doing so we shall conclude the proof of all of our new results (modulo Theorem 1.6), including the lower bound on the integrality gap for the Goemans–Linial SDP.

1.5. From isoperimetry to non-embeddability

An equivalent formulation of Theorem 1.6 is that every finitely supported function $\upphi:\mathbb{Z}^{5}\to L_{1}$ satisfies the following Poincaré-type inequality.

[TABLE]

Indeed, Theorem (1.6) is nothing more than the special case $\upphi=\mathbf{1}_{\Omega}$ of (12). Conversely, the fact that (12) follows from Theorem (1.6) is a straightforward application of the cut-cone representation of $L_{1}$ metrics (see e.g. (Deza and Laurent, 1997, Proposition 4.2.2) or (Naor, 2010, Corollary 3.2)), though our proof will yield the (seemingly) stronger statement (12) directly. Next, Section 3.2 of (Lafforgue and Naor, 2014) shows that (12) formally implies its local counterpart, which asserts that there exists a universal constant $\upalpha\geqslant 1$ such that for every $n\in\mathbb{N}$ and every $\upphi:\mathbb{Z}^{5}\to L_{1}$ we have

[TABLE]

To deduce Theorem 1.3, suppose that $R\geqslant 2$, that $\upomega:[0,\infty)\to[0,\infty)$ is nondecreasing and that the mapping $\upphi:\mathcal{B}_{R}\to L_{1}$ satisfies (5). For notational convenience, fix two universal constants $\upbeta\in(0,1)$ and $\upgamma\in(1,\infty)$ such that $\upbeta\sqrt{t}\leqslant d_{W}(Z^{t},\mathbf{0})\leqslant\upgamma\sqrt{t}$ for every $t\in\mathbb{N}$. Note that (5) implies in particular that $\upomega(R)\lesssim R$, so for every $c\in(0,1)$ the left hand side of (6) is at most a universal constant multiple of $R^{2}$. Hence, it suffices to prove Theorem 1.3 when $R\geqslant 1+\max\{\upalpha,\upgamma\}$, where $\upalpha$ is the universal constant in (13). Denote $n=\lfloor\min\{R/(1+\upgamma),(R-1)/\upalpha\}\rfloor\in\mathbb{N}$. If $t\in\{1,\ldots,n^{2}\}$ and $h\in\mathcal{B}_{n}$ then $d_{W}(hZ^{t},\mathbf{0})\leqslant n+\upgamma\sqrt{t}\leqslant(1+\upgamma)n\leqslant R$, and therefore we may apply (5) with $x=hZ^{t}$ and $y=h$ to deduce that $\|\upphi(hZ^{t})-\upphi(h)\|_{1}\gtrsim\upomega(d_{W}(Z^{t},\mathbf{0}))\geqslant\upomega(\upbeta\sqrt{t})$. Consequently,

[TABLE]

where the second inequality in (1.5) uses the fact that $\upomega$ is non-decreasing, the penultimate step of (1.5) uses the change of variable $s=\upbeta\sqrt{u/2}$, and for the final step of (1.5) recall that $\upbeta<1$ and the definition of $n$. At the same time, by our choice of $n$ we have $h\upsigma\in\mathcal{B}_{\upalpha n+1}\subseteq\mathcal{B}_{R}$ for every $h\in\mathcal{B}_{\upalpha n}$ and $\upsigma\in S$, and so by (5) we have $\|\upphi(h\upsigma)-\upphi(h)\|_{1}\leqslant d_{W}(h\upsigma,h)=1$. The right hand side of (13) is therefore at most a universal constant multiple of $|\mathcal{B}_{\upalpha n}|\cdot|S|\lesssim(\upalpha n)^{6}\lesssim R^{6}$. By contrasting (1.5) with (13) we obtain that the desired estimate (6) indeed holds true.

1.6. Overview of the proof of Theorem 1.6

Our proof of (12), and hence also of Theorem 1.6, is carried out in a continuous setting that is equivalent to its discrete counterpart. Such a passage from continuous to discrete is commonplace, and in the present setting this was carried out in (Austin et al., 2013; Lafforgue and Naor, 2014). The idea is to consider a continuous group that contains $\mathbb{H}_{\mathbb{Z}}^{5}$ and to deduce the discrete inequality (12) from its (appropriately formulated) continuous counterpart via a partition of unity argument. There is an obvious way to embed $\mathbb{H}_{\mathbb{Z}}^{5}$ in a continuous group, namely by considering the same group of matrices as in (2), but with the entries $\mathsf{a},\mathsf{b},{\mathsf{c}},\mathsf{d},\mathsf{e}$ now allowed to be arbitrary real numbers instead of integers. This is a indeed a viable route and the ensuing discussion could be carried out by considering the resulting continuous matrix group. Nevertheless, it is notationally advantageous to work with a different (standard) realization of $\mathbb{H}_{\mathbb{Z}}^{5}$ which is isomorphic to the one that we considered thus far. We shall now introduce the relevant notation.

Fix an orthonormal basis $\{X_{1},X_{2},Y_{1},Y_{2},Z\}$ of $\mathbb{R}^{5}$. If $h=\upalpha_{1}X_{1}+\upalpha_{2}X_{2}+\upbeta_{1}Y_{1}+\upbeta_{2}Y_{2}+\upgamma Z\in\mathbb{R}^{5}$ then denote $x_{i}(h)=\upalpha_{i}$, $y_{i}(h)=\upbeta_{i}$ for $i\in\{1,2\}$ and $z(h)=\upgamma$, i.e., $x_{1},x_{2},y_{1},y_{2},z:\mathbb{R}^{5}\to\mathbb{R}$ are the coordinate functions corresponding to the above basis. The continuous Heisenberg group $\mathbb{H}^{5}$ is defined to be $\mathbb{R}^{5}$, equipped with the following group law.

[TABLE]

The identity element of $\mathbb{H}^{5}$ is $\mathbf{0}\in\mathbb{R}^{5}$ and the inverse of $h\in\mathbb{R}^{5}$ under the group law (15) is equal to $-h$. By directly computing Jacobians, one checks that the Lebesgue measure on $\mathbb{R}^{5}$ is invariant under the group operation given in (15), i.e., it is a Haar measure of $\mathbb{H}^{5}$. In what follows, in order to avoid confusing multiplication by scalars with the group law of $\mathbb{H}^{5}$, for every $h\in\mathbb{H}^{5}$ and $t\in\mathbb{R}$ we shall use the exponential notation $h^{t}=(th_{1},\ldots,th_{5})$; this agrees with the group law when $t\in\mathbb{Z}$. (This convention is not strictly necessary, but without it the ensuing discussion could become somewhat notationally confusing.)

The subgroup of $\mathbb{H}^{5}$ that is generated by $\{X_{1},X_{2},Y_{1},Y_{2}\}$ is the discrete Heisenberg group of dimension $5$, denoted $\mathbb{H}_{\mathbb{Z}}^{5}$. The apparent inconsistency with (2) is not an actual issue because it is straightforward to check that the two groups in question are in fact isomorphic. The linear span of $\{X_{1},Y_{1},Z\}$ is a subgroup of $\mathbb{H}^{5}$ which is denoted $\mathbb{H}^{3}$ (the $3$-dimensional Heisenberg group).

There is a canonical left-invariant metric on $\mathbb{H}^{5}$, commonly called the Carnot–Carathéodory metric, which we denote by $d$. We refer to (Capogna et al., 2007) for a precise definition of this metric. For the purpose of the present discussion it suffices to know that $d$ possesses the following properties. Firstly, for every $g,h\in\mathbb{H}^{5}$ and $\uptheta\in\mathbb{R}$ we have $d(\mathfrak{s}_{\uptheta}(g),\mathfrak{s}_{\uptheta}(h))=|\uptheta|d(g,h)$. Here, $\mathfrak{s}_{\uptheta}$ denotes the Heisenberg scaling by $\uptheta$, given by the formula

[TABLE]

Secondly, the restriction of $d$ to the subgroup $\mathbb{H}_{\mathbb{Z}}^{5}$ is bi-Lipschitz to the word metric induced by its generating set $\{X_{1}^{\pm 1},X_{2}^{\pm 1},Y_{1}^{\pm 1},Y_{2}^{\pm 1}\}$. Thirdly, there exists $C\in(1,\infty)$ such that every $h\in\mathbb{H}^{5}$ satisfies

[TABLE]

Given $r\in(0,\infty)$ we shall denote by $B_{r}\subseteq\mathbb{H}^{5}$ the open ball in the metric $d$ of radius $r$ centered at the identity element, i.e., $B_{r}=\{h\in\mathbb{H}^{5}:\ d(\mathbf{0},h)<r\}$. For $\Omega\subseteq\mathbb{H}^{5}$ the Lipschitz constant of a mapping $f:\Omega\to\mathbb{R}$ relative to the metric $d$ will be denoted by $\|f\|_{\operatorname{Lip}(\Omega)}$. For $s\in(0,\infty)$, the notation $\mathcal{H}^{s}$ will be used exclusively to denote the $s$-dimensional Hausdorff measure that is induced by the metric $d$ (see e.g. (Mattila, 1995)). One checks that $\mathcal{H}^{6}$ is proportional to the Lebesgue measure on $\mathbb{R}^{5}$ and that the restriction of $\mathcal{H}^{4}$ to the subgroup $\mathbb{H}^{3}$ is proportional to the Lebesgue measure on $\mathbb{H}^{3}$ (under the canonical identification of $\mathbb{H}^{3}$ with $\mathbb{R}^{3}$). For two measurable subsets $E,U\subseteq\mathbb{H}^{5}$ define the normalized vertical perimeter of $E$ in $U$ to be the function $\overline{\mathsf{v}}_{U}(E):\mathbb{R}\to[0,\infty]$ given by setting for every $s\in\mathbb{R}$,

[TABLE]

where $A\mathop{\triangle}B\stackrel{{\scriptstyle\mathrm{def}}}{{=}}(A\smallsetminus B)\cup(B\smallsetminus A)$ is the symmetric difference. We also denote $\overline{\mathsf{v}}(E)\stackrel{{\scriptstyle\mathrm{def}}}{{=}}\overline{\mathsf{v}}_{\mathbb{H}^{5}}(E)$.

The isoperimetric-type inequality of Theorem 1.7 below implies Theorem 1.6. See (Naor and Young, 2017) for an explanation of this (standard) deduction; the argument is a straightforward use of the co-area formula (see e.g. (Ambrosio, 2001; Magnani, 2011)) to pass from sets to functions, followed by the partition of unity argument of (Lafforgue and Naor, 2014, Section 3.3) to pass from the continuous setting to the desired discrete inequality (12).

Theorem 1.7.

\big{\|}\overline{\mathsf{v}}(E)\big{\|}_{L_{2}(\mathbb{R})}\lesssim\mathcal{H}^{5}(\partial E)* for all open $E\subseteq\mathbb{H}^{5}$.*

We shall now explain the overall strategy and main ideas of our proof of Theorem 1.7. Complete technical details are included in (Naor and Young, 2017). A key new ingredient appears in Section 1.6.1 below, which is the only place in our proof where we use the fact that we are dealing with $\mathbb{H}^{5}$ rather than $\mathbb{H}^{3}$. In fact, the analogue of Theorem 1.7 for $\mathbb{H}^{3}$ (i.e., with $\mathcal{H}^{5}(\partial E)$ replaced by $\mathcal{H}^{3}(\partial E)$ and $\overline{\mathsf{v}}(E)(\cdot)$ defined in the same way as in (1.6) but with $\mathcal{H}^{6}$ replaced by the restriction of $\mathcal{H}^{4}$ to $\mathbb{H}^{3}$) is false (see Section 1.7.1 below). The crux of the matter is the special case of Theorem 1.7 where the boundary of $E$ is (a piece of) an intrinsic Lipschitz graph. Such sets were introduced by Franchi, Serapioni, and Serra Cassano (Franchi et al., 2006). These sets can be quite complicated, and in particular they are not the same as graphs of functions (in the usual sense) that are Lipschitz with respect to the Carnot–Carathéodory metric. Our proof of this special case relies crucially on an $L_{2}$-variant of (12) for $\mathbb{H}^{3}$ that was proven in (Austin et al., 2013) using representation theory and in (Lafforgue and Naor, 2014) using Littlewood–Paley theory. In essence, our argument “lifts” a certain $L_{2}$ inequality in lower dimensions to a formally stronger endpoint $L_{1}$ (or isoperimetric-type) inequality in higher dimensions. Once the special case is established, we prove Theorem 1.7 in its full generality by decomposing an open set $E$ into parts whose boundaries are close to pieces of intrinsic Lipschitz graphs and applying the special case to each part of this decomposition. We deduce the desired estimate by summing up all the inequalities thus obtained. Such a “corona decomposition” is an important and widely-used tool in harmonic analysis on $\mathbb{R}^{n}$ that was formulated by David and Semmes in (David and Semmes, 1991). For the present purpose we need to devise an “intrinsic version” of a corona decomposition on the Heisenberg group. This step uses a different “coercive quantity” to control local overlaps, but for the most part it follows the lines of the well-understood methodology of David and Semmes, as described in the monographs (David and Semmes, 1991, 1993).

1.6.1. Intrinsic Lipschitz graphs

Set $V\stackrel{{\scriptstyle\mathrm{def}}}{{=}}\{h\in\mathbb{H}^{5}:\ x_{2}(h)=0\}$. For $f:V\to\mathbb{R}$ define

[TABLE]

where (18) uses the identification of $\mathsf{a}X_{1}+\mathsf{b}X_{2}+{\mathsf{c}}Y_{1}+\mathsf{d}Y_{2}+\mathsf{e}Z\in\mathbb{H}^{5}$ with $(\mathsf{a},\mathsf{b},{\mathsf{c}},\mathsf{d},\mathsf{e})\in\mathbb{R}^{5}$ and the identification of $(\mathsf{a},0,{\mathsf{c}},\mathsf{d},\mathsf{e})\in V$ with $(\mathsf{a},{\mathsf{c}},\mathsf{d},\mathsf{e})\in\mathbb{R}^{4}$ (thus we think of the domain of $f$ as equal to $\mathbb{R}^{4}$). The set $\Gamma_{f}$ is a typical intrinsic graph in $\mathbb{H}^{5}$. See (Naor and Young, 2017) for a discussion of the general case, which is equivalent to this case via a symmetry of $\mathbb{H}^{5}$ (so the ensuing discussion has no loss of generality). Suppose that $\uplambda\in(0,\infty)$. We say that $\Gamma_{f}$ is an intrinsic $\uplambda$-Lipschitz graph over the vertical hyperplane $V$ if

[TABLE]

Due to (18) the condition (19) amounts to a point-wise inequality for $f$ that is somewhat complicated, and in particular it does not imply that $f$ must be Lipschitz with respect to the restriction of the Carnot–Carathéodory metric to the hyperplane $V$, as explained in (Franchi et al., 2011, Remark 3.13).

Denote by $\Gamma_{f}^{+}=\{vX_{2}^{t}:\ v\in V\ \wedge\ t>f(v)\}$ the half-space that is bounded by the intrinsic graph $\Gamma_{f}$. Suppose that $\Gamma_{f}$ is an intrinsic $\uplambda$-Lipschitz graph with $\uplambda\in(0,1)$. We claim that

[TABLE]

When, say, $\uplambda\in(0,\frac{1}{2})$, the estimate (20) is in essence the special case of Theorem 1.7 for pieces of Lipschitz graphs. This is so because, due to the isoperimetric inequality for the Heisenberg group (Pansu, 1982), the right-hand side of (20) is at most a universal constant multiple of $\mathcal{H}^{5}(\partial(B_{r}\cap\Gamma_{f}^{+}))$ whenever $\mathcal{H}^{6}(B_{r}\cap\Gamma_{f}^{+})\gtrsim r^{6}$, i.e., provided that $\Gamma_{f}^{+}$ occupies a constant fraction of the volume of the ball $B_{r}$. The estimate (20) will be used below only in such a non-degenerate situation.

The advantage of working in $\mathbb{H}^{5}$ rather than $\mathbb{H}^{3}$ is that $V\subseteq\mathbb{H}^{5}$ can be sliced into copies of $\mathbb{H}^{3}$. We will bound $\|\overline{\mathsf{v}}_{B_{r}}(\Gamma^{+}_{f})\|_{L_{2}(\mathbb{R})}$ by decomposing $\Gamma_{f}^{+}$ into a corresponding family of slices. Write

[TABLE]

Recalling (15), one computes directly that $u=Y_{2}^{y_{2}(u)}h_{u}X_{2}^{x_{2}(u)}$. Let $C\in(1,\infty)$ be the universal constant in (16). A straightforward computation using (16) shows that $d(h_{u},\mathbf{0})\leqslant Cd(u,\mathbf{0})$. Also, (16) implies that $|y_{2}(u)|\leqslant Cd(u,\mathbf{0})$. These simple observations demonstrate that

[TABLE]

For every $\upchi\in\mathbb{R}$ define $f_{\upchi}:\mathbb{H}^{3}\to\mathbb{R}$ by $f_{\upchi}(h)=f(Y_{2}^{\upchi}h)$ (recall that $\mathbb{H}^{3}$ is the span of $\{X_{1},Y_{1},Z\}$, so $Y_{2}^{\upchi}h\in V$ is in the domain of $f$). Under this notation $u\in\Gamma^{+}_{f}$ if and only if $x_{2}(u)>f_{y_{2}(u)}(h_{u})$. Also, for every $\upalpha\in\mathbb{R}$ we have $uZ^{\upalpha}\in\Gamma^{+}_{f}$ if and only if $x_{2}(u)>f_{y_{2}(u)}(h_{u}Z^{\upalpha})$, since $h_{uZ^{\upalpha}}=h_{u}Z^{\upalpha}$ by (21). Due to (1.6) and (22), these observations imply that for every $s\in\mathbb{R}$ we have

[TABLE]

Recall that $\mathcal{H}^{6}$ is proportional to the Lebesgue measure on $\mathbb{H}^{5}$. Hence, if we continue to canonically identify $\mathsf{a}X_{1}+\mathsf{b}X_{2}+{\mathsf{c}}Y_{1}+\mathsf{d}Y_{2}+\mathsf{e}Z\in\mathbb{H}^{5}$ with $(\mathsf{a},\mathsf{b},{\mathsf{c}},\mathsf{d},\mathsf{e})\in\mathbb{R}^{5}$ and $\mathsf{a}X_{1}+{\mathsf{c}}Y_{1}+\mathsf{e}Z\in\mathbb{H}^{3}$ with $(\mathsf{a},{\mathsf{c}},\mathsf{e})\in\mathbb{R}^{3}$ then, recalling (21), the integral in the right hand side of (1.6.1) is proportional to

[TABLE]

where for each fixed $\mathsf{b},\mathsf{d}\in\mathbb{R}$ we made the change of variable $(\upalpha,\upgamma,\upepsilon)=(\mathsf{a},{\mathsf{c}},\mathsf{e}+\mathsf{b}\mathsf{d}/2)$. Since the restriction of the Hausdorff measure $\mathcal{H}^{4}$ to $\mathbb{H}^{3}$ is proportional to the Lebesgue measure on $\mathbb{H}^{3}\cong\mathbb{R}^{3}$, we conclude from the above considerations that for every $s\in\mathbb{R}$ we have

[TABLE]

Next, fix $h_{1},h_{2}\in\mathbb{H}^{3}$ and $\upchi\in\mathbb{R}$. Denote $w_{1}\stackrel{{\scriptstyle\mathrm{def}}}{{=}}Y_{2}^{\upchi}h_{1}X_{2}^{f_{\upchi}(h_{1})}$ and $w_{2}\stackrel{{\scriptstyle\mathrm{def}}}{{=}}Y_{2}^{\upchi}h_{2}X_{2}^{f_{\upchi}(h_{2})}$. By design we have $w_{1},w_{2}\in\Gamma_{f}$ and therefore we may apply (20) to deduce that

[TABLE]

where the first inequality in (1.6.1) uses (20) , the penultimate step of (1.6.1) uses the left-invariance of the metric $d$ and the fact that $X_{2}$ commutes with all of the elements of $\mathbb{H}^{3}$, and the final step of (1.6.1) uses (16) (twice). The estimate (1.6.1) simplifies to show that $|f_{\upchi}(h_{1})-f_{\upchi}(h_{2})|\lesssim d(h_{1},h_{2})/(1-\uplambda)$, i.e., for every fixed $\upchi\in\mathbb{R}$ the function $f_{\upchi}$ is Lipschitz on $\mathbb{H}^{3}$ with $\|f_{\upchi}\|_{\operatorname{Lip}(\mathbb{H}^{3})}\lesssim 1/(1-\uplambda)$.

In (Austin et al., 2013, Theorem 7.5) the following inequality was proved for a Lipschitz function $\uppsi:\mathbb{H}^{3}\to\mathbb{R}$ and $\uprho\in(0,\infty)$ as a consequence of a continuous $L_{2}$-variant of (12). Due to its quadratic nature, this variant can be proved using a decomposition into irreducible representations (i.e., a spectral argument).

[TABLE]

Consequently,

[TABLE]

In (27) we applied (26) with $\uppsi=f_{\upchi}$ for each $\upchi\in[-Cr,Cr]$, while using $\|f_{\upchi}\|_{\operatorname{Lip}(\mathbb{H}^{3})}\lesssim 1/(1-\uplambda)$. In (28) we made the change of variable $t=2^{2s}$. In (29) we used the Cauchy–Schwarz inequality while noting that $\mathcal{H}^{4}(B_{Cr}\cap\mathbb{H}^{3})\asymp r^{4}$. Finally, (30) follows from an application of (1.6.1). Now,

[TABLE]

where we estimated the second integral using the trivial bound $\overline{\mathsf{v}}_{B_{r}}(E)(s)\leqslant\mathcal{H}^{6}(B_{r})/2^{s}\asymp r^{6}/2^{s}$. By taking square roots of both sides of (1.6.1) we obtain the desired estimate (20). It is important to stress that this proof does not work for functions on $\mathbb{H}^{3}$ because it relies on slicing $\mathbb{H}^{5}$ into copies of $\mathbb{H}^{3}$. There is no analogue of (26) for $1$-dimensional vertical slices of $\mathbb{H}^{3}$.

1.6.2. An intrinsic corona decomposition

In Section 1.6.1 we presented the complete details of the proof of a crucial new ingredient that underlies the validity of Theorem 1.7. This ingredient is the only step that relies on a property of $\mathbb{H}^{5}$ that is not shared by $\mathbb{H}^{3}$. We believe that it is important to fully explain this key ingredient within this extended abstract, but this means that we must defer the details of the formal derivation of Theorem 1.7 from its special case (20) to the full version (Naor and Young, 2017). The complete derivation requires additional terminology and notation, but the main idea is to produce “intrinsic corona decompositions” in the Heisenberg group. Corona decompositions are an established tool in analysis for reducing the study of certain singular integrals on $\mathbb{R}^{n}$ to the case of Lipschitz graphs, starting with seminal works of David (David, 1984, 1991a) and Jones (Jones, 1989, 1990) on the Cauchy integral and culminating with the David–Semmes theory of quantitative rectifiability (David and Semmes, 1991, 1993). Our adaptation of this technique is mostly technical, but it will also involve a conceptually new ingredient, namely the use of quantitative monotonicity for this purpose. We will now outline the remainder of the proof of Theorem 1.7.

Our arguments hold for Heisenberg groups of any dimension (including $\mathbb{H}^{3}$), but we avoid introducing new notation by continuing to work with $\mathbb{H}^{5}$ for now. The first step is to show that in order to establish Theorem 1.7 it suffices to prove that for every $r\in(0,\infty)$ and every $E\subseteq\mathbb{H}^{5}$, we have $\|\overline{\mathsf{v}}_{B_{r}}(E)\|_{L_{2}(\mathbb{R})}\lesssim r^{5}$ under the additional assumption that the sets $E$, $\mathbb{H}^{5}\smallsetminus E$, and $\partial E$ are $r$-locally Ahlfors-regular, i.e., $\mathcal{H}^{6}(uB_{\uprho}\cap E)\asymp\uprho^{6}\asymp\mathcal{H}^{6}(vB_{\uprho}\smallsetminus E)$ and $\mathcal{H}^{5}(wB_{\uprho}\cap\partial E)\asymp\uprho^{5}$ for all $\uprho\in(0,r)$ and $(u,v,w)\in E\times(\mathbb{H}^{5}\smallsetminus E)\times\partial E$. We prove this by first applying a Heisenberg scaling and an approximation argument to reduce Theorem 1.7 to the case that $E$ is a “cellular set,” i.e., it is a union of parallelepipeds of the form $h[-\frac{1}{2},\frac{1}{2}]^{5}$ as $h$ ranges over a subset of the discrete Heisenberg group $\mathbb{H}_{\mathbb{Z}}^{5}\subseteq\mathbb{H}^{5}$. Any such set is Ahlfors-regular on sufficiently small balls. We next argue that $E$ can be decomposed into sets that satisfy the desired local Ahlfors-regularity. The full construction of this decomposition appears in (Naor and Young, 2017), but we remark briefly that it amounts to the following natural “greedy” iterative procedure. If one of the sets $E,\mathbb{H}^{5}\smallsetminus E,\partial E$ were not locally Ahlfors-regular then there would be some smallest ball $B$ such that the density of $E$, $\mathbb{H}^{5}\smallsetminus E$ or $\partial E$ is either too low or too high on $B$. By replacing $E$ by either $E\cup B$ or $E\smallsetminus B$, we cut off a piece of $\partial E$ and decrease $\mathcal{H}^{5}(\partial E)$. Since $B$ was the smallest ball where Ahlfors-regularity fails, $E,\mathbb{H}^{5}\smallsetminus E,\partial E$ are Ahlfors-regular on balls smaller than $B$. Repeating this process eventually reduces $E$ to the empty set, and we arrive at the conclusion of Theorem 1.7 for the initial set $E$ by proving the (local version of) the theorem for each piece of this decomposition, then summing the resulting inequalities. We will therefore suppose from now on that $E$, $\mathbb{H}^{5}\smallsetminus E$ and $\partial E$ are all locally Ahlfors-regular.

The next step is the heart of the matter: approximating $\partial E$ by intrinsic Lipschitz graphs so that we can use the fact that Theorem 1.7 holds for (pieces of) such graphs. The natural way to do this is to construct (an appropriate Heisenberg version of) a corona decomposition in the sense of (David and Semmes, 1991, 1993). Such a decomposition covers $\partial E$ by two types of sets, called stopping-time regions and bad cubes. Stopping-time regions correspond to parts of $\partial E$ that are close to intrinsic Lipschitz graphs, and bad cubes correspond to parts of $\partial E$, like sharp corners, that are not. The multiplicity of this cover depends on the shape of $\partial E$ at different scales. For example, $\partial E$ might look smooth on a large neighborhood of a point $x$, jagged at a medium scale, then smooth again at a small scale. If so, then $x$ is contained in a large stopping-time region, a medium-sized bad cube, and a second small stopping-time region. A cover like this is a corona decomposition if it satisfies a Carleson packing condition (see (Naor and Young, 2017)) that bounds its average multiplicity on any ball.

We construct our cover following the well-established methods of (David and Semmes, 1991, 1993). We start by constructing a sequence of nested partitions of $\partial E$ into pieces called cubes; this is a standard construction due to Christ (Christ, 1990) and David (David, 1991b) and only uses the Ahlfors regularity of $\partial E$. These partitions are analogues of the standard tilings of $\mathbb{R}^{n}$ into dyadic cubes. Next, we classify the cubes into good cubes, which are close to a piece of a hyperplane, and bad cubes, which are not. In order to produce a corona decomposition, there cannot be too many bad cubes, i.e., they must satisfy a Carleson packing condition. In (David and Semmes, 1991, 1993), this condition follows from quantitative rectifiability; the surface in question is assumed to satisfy a condition that bounds the sum of its (appropriately normalized) local deviations from hyperplanes. These local deviations are higher-dimensional versions of Jones’ $\upbeta$-numbers (Jones, 1989, 1990), and the quantitative rectifiability assumption leads to the desired packing condition. In the present setting, the packing condition follows instead from quantitative non-monotonicity. The concept of the quantitative non-monotonicity of a set $E\subseteq\mathbb{H}^{5}$ (see (Naor and Young, 2017)) was first defined in (Cheeger et al., 2009, 2011), where the kinematic formula for the Heisenberg group was used to show that the total non-monotonicity of all of the cubes is at most a constant multiple of $\mathcal{H}^{5}(\partial E)$. This means that there cannot be many cubes that have large non-monotonicity. By a result of (Cheeger et al., 2009, 2011), if a set has small non-monotonicity, then its boundary is close to a hyperplane. Consequently, most cubes are close to hyperplanes and are therefore good. (The result in (Cheeger et al., 2009, 2011) is stronger than what we need for this proof; it provides power-type bounds on how closely a nearly-monotone surface approximates a hyperplane. For our purposes, it is enough to have some bound (not necessarily power-type) on the shape of nearly-monotone surfaces, and we can deduce the bound that we need by applying a quick compactness argument to a result from (Cheeger and Kleiner, 2010b) that states that if a set is precisely monotone (i.e., every line intersects its boundary in at most one point), then it is a half-space.)

Next, we partition the good cubes into stopping-time regions by using an iterative construction that corrects overpartitioning that may have occurred when the Christ cubes were constructed. If $Q$ is a largest good cube that hasn’t been treated yet and if $P$ is its approximating half-space, we find all of the descendants of $Q$ with approximating half-spaces that are sufficiently close to $P$. If we glue these half-spaces together using a partition of unity, the result is an intrinsic Lipschitz half-space that approximates all of these descendants. By repeating this procedure for each untreated cube, we obtain a collection of stopping-time regions. These regions satisfy a Carleson packing condition because if a point $x\in\partial E$ is contained in many different stopping-time regions, then either $x$ is contained in many different bad cubes, or $x$ is contained in good cubes whose approximating hyperplanes point in many different directions. In either case, these cubes generate non-monotonicity, so there can only be a few points with large multiplicity.

The construction above leads to the proof of Theorem 1.7 as follows. The vertical perimeter of $\partial E$ comes from three sources: the bad cubes, the approximating Lipschitz graphs, and the error incurred by approximating a stopping-time region by an intrinsic Lipschitz graph. By the Carleson packing condition, there are few bad cubes, and they contribute vertical perimeter on the order of $\mathcal{H}^{5}(\partial E)$. By the result of Section 1.6.1, the intrinsic Lipschitz graphs also contribute vertical perimeter on the order of $\mathcal{H}^{5}(\partial E)$. Finally, the vertical perimeter of the difference between a stopping-time region and an intrinsic Lipschitz graph is bounded by the size of the stopping-time region. The stopping-time regions also satisfy a Carleson packing condition, so these errors also contribute vertical perimeter on the order of $\mathcal{H}^{5}(\partial E)$. Summing these contributions, we obtain the desired bound.

1.7. Historical overview and directions for further research

Among the well-established deep and multifaceted connections between theoretical computer science and pure mathematics, the Sparsest Cut Problem stands out for its profound and often unexpected impact on a variety of areas. Indeed, previous research on this question came hand-in-hand with the development of remarkable mathematical and algorithmic ideas that spurred many further works of importance in their own right. Because the present work belongs to this tradition, we will try to put it into context by elaborating further on the history of these investigations and describing directions for further research and open problems. Some of these directions will appear in forthcoming work.

The first polynomial-time algorithm for Sparsest Cut with approximation ratio $O(\log n)$ was obtained in the important work (Leighton and Rao, 1999), which studied the notable special case of Sparsest Cut with Uniform Demands (see Section 1.7.3 below). This work introduced a linear programming relaxation and developed influential techniques for its analysis, and it has led to a myriad of algorithmic applications. The seminal contributions (Linial et al., 1995; Aumann and Rabani, 1998) obtained the upper bound $\uprho_{\mathsf{GL}}(n)\lesssim\log n$ in full generality by incorporating a classical embedding theorem of Bourgain (Bourgain, 1985), thus heralding the transformative use of metric embeddings in algorithm design. The matching lower bound on the integrality gap of this linear program was proven in (Leighton and Rao, 1999; Linial et al., 1995). This showed for the first time that Bourgain’s embedding theorem is asymptotically sharp and was the first demonstration of the power of expander graphs in the study of metric embeddings.

A $O(\sqrt{\log n})$ upper bound for the approximation ratio of the Goemans–Linial algorithm in the case of uniform demands was obtained in the important work (Arora et al., 2009). This work relied on a clever use of the concentration of measure phenomenon and introduced influential techniques such as a “chaining argument” for metrics of negative type and the use of expander flows. (Arora et al., 2009) also had direct impact on results in pure mathematics, including combinatorics and metric geometry; see e.g. the “edge replacement theorem” and the estimates on the observable diameter of doubling metric measure spaces in (Naor et al., 2005). The best-known upper bound $\uprho_{\mathsf{GL}}(n)\lesssim(\log n)^{\frac{1}{2}+o(1)}$ of (Arora et al., 2008) built on the (then very recent) development of two techniques: The chaining argument of (Arora et al., 2009) (through its refined analysis in (Lee, 2005)) and the measured descent embedding method of (Krauthgamer et al., 2005) (through its statement as a gluing technique for Lipschitz maps in (Lee, 2005)). Another important input to (Arora et al., 2008) was a re-weighting argument of (Chawla et al., 2008) that allowed for the construction of an appropriate “random zero set” from the argument of (Arora et al., 2009; Lee, 2005) (see (Naor, 2010, 2014) for more on this notion and its significance).

The impossibility result (Khot and Vishnoi, 2015) that refuted the Goemans–Linial conjecture relied on a striking link to complexity theory through the Unique Games Conjecture (UGC), as well as an interesting use of discrete harmonic analysis (through (Bourgain, 2002)) in this context; see also (Krauthgamer and Rabani, 2009) for an incorporation of a different tool from discrete harmonic analysis (namely (Kahn et al., 1988), following (Khot and Naor, 2006)) for the same purpose, as well as (Chawla et al., 2006; Chuzhoy and Khanna, 2007) for computational hardness. The best impossibility result currently known (Kane and Meka, 2013) for Sparsest Cut with Uniform Demands relies on the development of new pseudorandom generators.

The idea of using the geometry of the Heisenberg group to bound $\uprho_{\mathsf{GL}}(n)$ from below originated in (Lee and Naor, 2006), where the relevant metric of negative type was constructed through a complex-analytic argument, and initial (qualitative) impossibility results were presented through the use of Pansu’s differentiation theorem (Pansu, 1989) and the Radon–Nikodým Property from functional analysis (see e.g. (Benyamini and Lindenstrauss, 2000)). In (Cheeger and Kleiner, 2010a), it was shown that the Heisenberg group indeed provides a proof that $\lim_{n\to\infty}\uprho_{\mathsf{GL}}(n)=\infty$. This proof introduced a remarkable new notion of differentiation for $L_{1}$-valued mappings, which led to the use of tools from geometric measure theory (Franchi et al., 2001, 2003) to study the problem. A different proof that $\mathbb{H}^{3}$ fails to admit a bi-Lipschitz embedding into $L_{1}$ was found in (Cheeger and Kleiner, 2010b), where a classical notion of metric differentiation (Kirchheim, 1994) was used in conjunction with the novel idea to consider monotonicity of sets in this context, combined with a sub-Riemannian-geometric argument that yielded a classification of monotone subsets of $\mathbb{H}^{3}$. The main result of (Cheeger et al., 2011) finds a quantitative lower estimate for the scale at which this differentiation argument can be applied, leading to a lower bound of $(\log n)^{\Omega(1)}$ on $\uprho_{\mathsf{GL}}(n)$. This result relies on a mixture of the methods of (Cheeger and Kleiner, 2010a) and (Cheeger and Kleiner, 2010b) and requires overcoming obstacles that are not present in the original qualitative investigation. In particular, (Cheeger et al., 2011) introduced the quantitative measures of non-monotonicity that we use in the present work to find crucial bounds in the construction of an intrinsic corona decomposition. The quantitative differentiation bound of (Cheeger et al., 2011) remains the best bound currently known, and it would be very interesting to discover the sharp behavior in this more subtle question.

The desire to avoid the (often difficult) need to obtain sharp bounds for quantitative differentiation motivated the investigations in (Austin et al., 2013; Lafforgue and Naor, 2014). In particular, (Austin et al., 2013) devised a method to prove sharp (up to lower order factors) nonembeddability statements for the Heisenberg group based on a cohomological argument and a quantitative ergodic theorem. For Hilbert-space valued mappings, (Austin et al., 2013) used a cohomological argument in combination with representation theory to prove the following quadratic inequality for every finitely supported function $\upphi:\mathbb{H}_{Z}^{5}\to L_{2}$.

[TABLE]

In (Lafforgue and Naor, 2014) a different approach based on Littlewood–Paley theory was devised, leading to the following generalization of (32) that holds true for every $p\in(1,2]$ and every finitely supported $\upphi:\mathbb{H}^{5}\to L_{p}$.

[TABLE]

for some $C(p)\in(0,\infty)$. See (Lafforgue and Naor, 2014) for a strengthening of (33) that holds for general uniformly convex targets (using the recently established (Martínez et al., 2006) vector-valued Littlewood–Paley–Stein theory for the Poisson semigroup). These functional inequalities yield sharp non-embeddability estimates for balls in $\mathbb{H}^{5}_{\mathbb{Z}}$, but the method of (Lafforgue and Naor, 2014) inherently yields a constant $C(p)$ in (33) that satisfies $\lim_{p\to 1}C(p)=\infty$. The estimate (13) that we prove here for $L_{1}$-valued mappings is an endpoint estimate corresponding to (33), showing that the best possible $C(p)$ actually remains bounded as $p\to 1$. This confirms a conjecture of (Lafforgue and Naor, 2014) and is crucial for the results that we obtain here.

As explained in Section 1.6.1, our proof of (32) uses the $\mathbb{H}^{3}$-analogue of (32). It should be mentioned at this juncture that the proofs of (32) and (33) in (Austin et al., 2013; Lafforgue and Naor, 2014) were oblivious to the dimension of the underlying Heisenberg group.333Thus far in this extended abstract we recalled the definitions of $\mathbb{H}^{5}$ and $\mathbb{H}^{3}$ but not of higher-dimensional Heisenberg groups (since they are not needed for any of the applications that are obtain here). Nevertheless, it is obvious how to generalize either the matrix group or the group modelled on $\mathbb{R}^{5}$ that we considered above to obtain the Heisenberg group $\mathbb{H}^{2k+1}$ for any $k\in\mathbb{N}$. An unexpected aspect of the present work is that the underlying dimension does play a role at the endpoint $p=1$, with the analogue of (13) (or Theorem 1.6) for $\mathbb{H}^{3}$ being in fact incorrect; see Section 1.7.1 below. In the full version (Naor and Young, 2017) of this paper we shall establish the $\mathbb{H}^{2k+1}$-analogue of Theorem 1.6 for every $k\in\{2,3,\ldots\}$, in which case the implicit constant depends on $k$, and we shall also obtain the sharp asymptotic behavior as $k\to\infty$.

As we recalled above, past progress on the Sparsest Cut Problem came hand-in-hand with meaningful mathematical developments. The present work is a culmination of a long-term project that is rooted in mathematical phenomena that are interesting not just for their relevance to approximation algorithms but also for their connections to the broader mathematical world. In the ensuing subsections we shall describe some further results and questions related to this general direction.

1.7.1. The $3$-dimensional Heisenberg group

The investigation of the possible validity of an appropriate analogue of Theorem 1.6 with $\mathbb{H}_{\mathbb{Z}}^{5}$ replaced by $\mathbb{H}_{\mathbb{Z}}^{3}$ remains an intriguing mystery and a subject of ongoing research that will be published elsewhere. This ongoing work shows that Theorem 1.6 fails for $\mathbb{H}_{\mathbb{Z}}^{3}$, but that there exists $p\in(2,\infty)$ such that for every $\Omega\subseteq\mathbb{H}_{\mathbb{Z}}^{3}$ we have

[TABLE]

A simple argument shows that $\sup_{s\in\mathbb{N}}|\partial^{s}_{\mathsf{v}}\Omega|/\sqrt{s}\leqslant\upgamma|\partial_{\mathsf{h}}\Omega|$ for some universal constant $\upgamma>0$. Hence, for every $t\in\mathbb{N}$ we have

[TABLE]

This implies that the left hand side of (34) is bounded from above by a universal constant multiple of $|\partial_{\mathsf{v}}\Omega|^{2/p}|\partial_{\mathsf{h}}\Omega|^{1-2/p}$. Therefore (34) is weaker than the estimate $|\partial_{\mathsf{v}}\Omega|\lesssim|\partial_{\mathsf{h}}\Omega|$ of Theorem 1.6. It would be interesting to determine the infimum over those $p$ for which (34) holds true for every $\Omega\subseteq\mathbb{H}_{\mathbb{Z}}^{3}$, with our ongoing work showing that it is at least $4$. In fact, this work shows that for every $R\geqslant 2$ the $L_{1}$ distortion of the ball of radius $R$ in $\mathbb{H}_{\mathbb{Z}}^{3}$ is at most a constant multiple of $\sqrt[4]{\log R}$ — asymptotically less than the distortion of the ball of the same radius in $\mathbb{H}_{\mathbb{Z}}^{5}$. It would be interesting to determine the correct asymptotics of this distortion, with the best-known lower bound remaining that of (Cheeger et al., 2011), i.e., a constant multiple of $(\log R)^{\updelta}$ for some universal constant $\updelta>0$. It should be stressed, however, that the algorithmic application of Theorem 1.6 that is obtained here uses Theorem (1.6) as stated for $\mathbb{H}_{\mathbb{Z}}^{5}$, and understanding the case of $\mathbb{H}_{\mathbb{Z}}^{3}$ would not yield any further improvement. So, while the above questions are geometrically and analytically interesting in their own right, they are not needed for applications that we currently have in mind.

1.7.2. Metric embeddings

Theorem 1.2 also yields a sharp result for the general problem of finding the asymptotically largest-possible $L_{1}$ distortion of a finite doubling metric space with $n$ points. A metric space $(X,d_{X})$ is said to be $K$-doubling for some $K\in\mathbb{N}$ if every ball in $X$ (centered anywhere and of any radius) can be covered by $K$ balls of half its radius. By (Krauthgamer et al., 2005),

[TABLE]

As noted in (Gupta et al., 2003), the dependence on $|X|$ in (35), but with a worse dependence on $K$, follows by combining results of (Assouad, 1983) and (Rao, 1999) (the dependence on $K$ that follows from (Assouad, 1983; Rao, 1999) was improved significantly in (Gupta et al., 2003)). The metric space $(\mathbb{Z}^{5},d_{W})$ is $O(1)$-doubling because $|\mathcal{B}_{R}|\asymp R^{6}$ for every $R\geqslant 1$. Theorem 1.2 shows that (35) is sharp when $K=O(1)$, thus improving over the previously best-known construction (Lee and Sidiropoulos, 2011) of arbitrarily large $O(1)$-doubling finite metric spaces $\{(X_{i},d_{i})\}_{i=1}^{\infty}$ for which $c_{1}(X_{i},d_{i})\gtrsim\sqrt{(\log|X_{i}|)/\log\log|X_{i}|}$. Probably (35) is sharp for every $K\leqslant|X|$; conceivably this could be proven by incorporating Theorem 1.2 into the argument of (Jaffe et al., 2011), but we shall not pursue this here. Theorem 1.2 establishes for the first time the existence of a metric space that simultaneously has several useful geometric properties and poor (indeed, worst possible) embeddability into $L_{1}$. By virtue of being $O(1)$-doubling, the metric space $(\mathbb{Z}^{5},d_{W})$ also has Markov type $2$ due to (Ding et al., 2013) (which improves over (Naor et al., 2006), where the conclusion that it has Markov type $p$ for every $p<2$ was obtained). For more on the bi-Lipschitz invariant Markov type and its applications, see (Ball, 1992; Naor, 2012). The property of having Markov type $2$ is shared by the construction of (Lee and Sidiropoulos, 2011), which is also $O(1)$-doubling, but $(\mathbb{Z}^{5},d_{W})$ has additional features that the example of (Lee and Sidiropoulos, 2011) fails to have. For one, it is a group; for another, by (Li, 2014, 2016) we know that $(\mathbb{Z}^{5},d_{W})$ has Markov convexity $4$ (and no less). (See (Lee et al., 2009; Mendel and Naor, 2013) for background on the bi-Lipschitz invariant Markov convexity and its consequences.) By (Mendel and Naor, 2013, Section 3) the example of (Lee and Sidiropoulos, 2011) does not have Markov convexity $p$ for any finite $p$. No examples of arbitrarily large finite metric spaces $\{(X_{i},d_{i})\}_{i=1}^{\infty}$ with bounded Markov convexity (and Markov convexity constants uniformly bounded) such that $c_{1}(X_{i},d_{i})\gtrsim\sqrt{\log|X_{i}|}$ were previously known to exist. Analogous statements are known to be impossible for Banach spaces (Milman and Wolfson, 1978), so it is natural in the context of the Ribe program (see the surveys (Naor, 2012; Ball, 2013) for more on this research program) to ask whether there is a potential metric version of (Milman and Wolfson, 1978); the above discussion shows that there is not.

1.7.3. The Sparsest Cut Problem with Uniform Demands

An important special case of the Sparsest Cut Problem is when the demand matrix $D$ is the matrix $\mathbf{1}_{\{1,\ldots,n\}\times\{1,\ldots,n\}}\in M_{n}(\mathbb{R})$ all of whose entries equal $1$ and the capacity matrix $C$ lies in $M_{n}(\{0,1\})$, i.e., all its entries are either [math] or $1$. This is known as the Sparsest Cut Problem with Uniform Demands. In this case $C$ can also be described as the adjacency matrix of a graph $G$ whose vertex set is $\{1,\ldots,n\}$ and whose edge set consists of those unordered pairs $\{i,j\}\subseteq\{1,\ldots,n\}$ for which $C_{ij}=1$. With this interpretation, given $A\subseteq\{1,\ldots,n\}$ the numerator in (1) equals twice the number of edges that are incident to $A$ in $G$. And, since $D=\mathbf{1}_{\{1,\ldots,n\}\times\{1,\ldots,n\}}$, the denominator in (1) is equal to $2|A|(n-|A|)\asymp n\min\{|A|,|\{1,\ldots,n\}\smallsetminus A|\}$. So, the Sparsest Cut Problem with Uniform Demands asks for an algorithm that takes as input a finite graph and outputs a quantity which is bounded above and below by universal constant multiples of its conductance (Sinclair and Jerrum, 1989) divided by $n$. The Goemans–Linial integrality gap corresponding to this special case is

[TABLE]

The Goemans–Linial algorithm furnishes the best-known approximation ratio also in the case of uniform demands. By the important work (Arora et al., 2009) we have $\uprho_{\mathsf{GL}}^{\mathrm{unif}}(n)\lesssim\sqrt{\log n}$, improving over the previous bound $\uprho_{\mathsf{GL}}^{\mathrm{unif}}(n)\lesssim\log n$ of (Leighton and Rao, 1999). As explained in (Cheeger et al., 2009), the present approach based on (fixed dimensional) Heisenberg groups cannot yield a lower bound on $\uprho_{\mathsf{GL}}^{\mathrm{unif}}(n)$ that tends to $\infty$ with $n$. The best-known lower bound (Kane and Meka, 2013) is $\uprho_{\mathsf{GL}}^{\mathrm{unif}}(n)\geqslant\exp(c\sqrt{\log\log n})$ for some universal constant $c>0$, improving over the previous bound $\uprho_{\mathsf{GL}}^{\mathrm{unif}}(n)\gtrsim\log\log n$ of (Devanur et al., 2006). Determining the asymptotic behavior of $\uprho_{\mathsf{GL}}^{\mathrm{unif}}(n)$ remains an intriguing open problem.