We consider the sparsification of sums F : ℝn → ℝ+ where F(x) = f1(⟨ a1,x⟩) + ⋯ + fm(⟨ am,x⟩) for vectors a1,…,am ∈ ℝn and functions f1,…,fm : ℝ → ℝ+. We show that (1+ε)-approximate sparsifiers of F with support size n/ε2 (logn/ε)O(1) exist whenever the functions f1,…,fm are symmetric, monotone, and satisfy natural growth bounds. Additionally, we give efficient algorithms to compute such a sparsifier assuming each fi can be evaluated efficiently. Our results generalize the classical case of ℓp sparsification, where fi(z) = |z|p, for p ∈ (0, 2], and give the first near-linear size sparsifiers in the well-studied setting of the Huber loss function and its generalizations, e.g., fi(z) = min{|z|p, |z|2} for 0 < p ≤ 2. Our sparsification algorithm can be applied to give near-optimal reductions for optimizing a variety of generalized linear models including ℓp regression for p ∈ (1, 2] to high accuracy, via solving (logn)O(1) sparse regression instances with m ≤ n(logn)O(1), plus runtime proportional to the number of nonzero entries in the vectors a1, …, am.

Abstract-For any norms $N_{1}, \ldots, N_{m}$ on $\mathbb{R}^{n}$ and $N(x):= N_{1}(x)+\cdots+N_{m}(x)$, we show there is a sparsified norm $\tilde{N}(x)= w_{1} N_{1}(x)+\cdots+w_{m} N_{m}(x)$ such that $|N(x)-\tilde{N}(x)| \leqslant \varepsilon N(x)$ for all $x \in \mathbb{R}^{n}$, where $w_{1}, \ldots, w_{m}$ are non-negative weights, of which only $O\left(\varepsilon^{-2} n \log (n / \varepsilon)(\log n)^{2.5}\right)$ are non-zero. Additionally, we show that such weights can be found with high probability in time $O\left(m(\log n)^{O(1)}+\right.$ poly $\left.(n)\right) T$, where T is the time required to evaluate a norm $N_{i}(x)$, assuming that $N(x)$ is poly $(n)$ equivalent to the Euclidean norm. This immediately yields analogous statements for sparsifying sums of symmetric submodular functions. More generally, we show how to sparsify sums of p th powers of norms when the sum is p-uniformly smooth.1

In a hypergraph on n vertices where D is the maximum size of a hyperedge, there is a weighted hypergraph spectral -sparsifier with at most O(−2 log(D) · n logn) hyperedges. This improves over the bound of Kapralov, Krauthgamer, Tardos and Yoshida (2021) who achieve O(−4 n (logn)3), as well as the bound O(−2 D3 n logn) obtained by Bansal, Svensson, and Trevisan (2019). The same sparsification result was obtained independently by Jambulapati, Liu, and Sidford (2022).

We present an $O((\log n)^2)$-competitive algorithm for metrical task systems (MTS) on any $n$-point metric space that is also $1$-competitive for service costs. This matches the competitive ratio achieved by Bubeck, Cohen, Lee, and Lee (2019) and the refined competitive ratios obtained by Coester and Lee (2019). Those algorithms work by first randomly embedding the metric space into an ultrametric and then solving MTS there. In contrast, our algorithm is cast as regularized gradient descent where the regularizer is a multiscale metric entropy defined directly on the metric space. This answers an open question of Bubeck (Highlights of Algorithms, 2019).

We consider metrical task systems on tree metrics and present an $O(\mathrm{depth} \times \log n)$-competitive randomized algorithm based on the mirror descent framework introduced in our prior work on the $k$-server problem. For the special case of hierarchically separated trees (HSTs), we use mirror descent to refine the standard approach based on gluing unfair metrical task systems. This yields an $O(\log n)$-competitive algorithm for HSTs, thus removing an extraneous $\log\log n$ in the bound of Fiat and Mendel (2003). Combined with well-known HST embedding theorems, this also gives an $O((\log n)^2)$-competitive randomized algorithm for every $n$-point metric space.

Recall the classical hypothesis testing setting with two sets of probability distributions P and Q. One receives either n i.i.d. samples from a distribution p E P or from a distribution q E Q and wants to decide from which set the points were sampled. It is known that the optimal exponential rate at which errors decrease can be achieved by a simple maximum-likelihood ratio test which does not depend on p or q, but only on the sets P and Q. We consider an adaptive generalization of this model where the choice of p E P and q E Q can change in each sample in some way that depends arbitrarily on the previous samples. In other words, in the kth round, an adversary, having observed all the previous samples in rounds 1, . . . , k - 1, chooses pk E P and qk E Q, with the goal of confusing the hypothesis test. We prove that even in this case, the optimal exponential error rate can be achieved by a simple maximum-likelihood test that depends only on P and Q. We then show that the adversarial model has applications in hypothesis testing for quantum states using restricted measurements. For example, it can be used to study the problem of distinguishing entangled states from the set of all separable states using only measurements that can be implemented with local operations and classical communication (LOCC). The basic idea is that in our setup, the deleterious effects of entanglement can be simulated by an adaptive classical adversary. We prove a quantum Stein's Lemma in this setting: In many circumstances, the optimal hypothesis testing rate is equal to an appropriate notion of quantum relative entropy between two states. In particular, our arguments yield an alternate proof of Li and Winter's recent strengthening of strong subadditivity for von Neumann entropy.

We show that on every $n$-point HST metric, there is a randomized online algorithm for metrical task systems (MTS) that is $1$-competitive for service costs and $O(\log n)$-competitive for movement costs. In general, these refined guarantees are optimal up to the implicit constant. While an $O(\log n)$-competitive algorithm for MTS on HST metrics was developed by Bubeck et al. (2018), that approach could only establish an $O((\log n)^2)$-competitive ratio when the service costs are required to be $O(1)$-competitive. Our algorithm is an instantiation of online mirror descent with the regularizer derived from a multiscale conditional entropy. In fact, our algorithm satisfies a set of even more refined guarantees; we are able to exploit this property to combine it with known random embedding theorems and obtain, for {\em any} $n$-point metric space, a randomized algorithm that is $1$-competitive for service costs and $O((\log n)^2)$-competitive for movement costs.

We consider metrical task systems on tree metrics, and present an O(depth×log n)-competitive randomized algorithm based on the mirror descent framework introduced in our prior work on the k-server problem. For the special case of hierarchically separated trees (HSTs), we use mirror descent to refine the standard approach based on gluing unfair metrical task systems. This yields an O(log n)-competitive algorithm for HSTs, thus removing an extraneous log log n in the bound of Fiat and Mendel (2003). Combined with well-known HST embedding theorems, this also gives an O((log n)2)-competitive randomized algorithm for every n-point metric space.

The relationship between the sparsest cut and the maximum concurrent multi-flow in graphs has been studied extensively. For general graphs, the worst-case gap between these two quantities is now settled: When there are k terminal pairs, the flow-cut gap is O(log k), and this is tight. But when topological restrictions are placed on the flow network, the situation is far less clear. In particular, it has been conjectured that the flow-cut gap in planar networks is O(1), while the known bounds place the gap somewhere between 2 (Lee and Raghavendra, 2003) and (Rao, 1999). A seminal result of Okamura and Seymour (1981) shows that when all the terminals of a planar network lie on a single face, the flow-cut gap is exactly 1. This setting can be generalized by considering planar networks where the terminals lie on one of γ > 1 faces in some fixed planar drawing. Lee and Sidiropoulos (2009) proved that the flow-cut gap is bounded by a function of γ, and Chekuri, Shepherd, and Weibel (2013) showed that the gap is at most 3γ. We significantly improve these asymptotics by establishing that the flow-cut gap is O(log γ). This is achieved by showing that the edge-weighted shortest-path metric induced on the terminals admits a stochastic embedding into trees with distortion O(log γ). The latter result is tight, e.g., for a square planar lattice on Θ(γ) vertices. The preceding results refer to the setting of edge-capacitated networks. For vertex-capacitated networks, it can be significantly more challenging to control flow-cut gaps. While there is no exact vertex-capacitated version of the Okamura-Seymour Theorem, an approximate version holds; Lee, Mendel, and Moharrami (2015) showed that the vertex-capacitated flow-cut gap is O(1) on planar networks whose terminals lie on a single face. We prove that the flow-cut gap is O(γ) for vertex-capacitated instances when the terminals lie on at most γ faces. In fact, this result holds in the more general setting of submodular vertex capacities.

We exhibit a poly(log k)-competitive randomized algorithm for the k-server problem on any metric space. The best previous result independent of the geometry of the underlying metric space is the 2k-1 competitive ratio established for the deterministic work function algorithm by Koutsoupias and Papadimitriou (1995). Even for the special case when the underlying metric space is the real line, the best known competitive ratio was k. Since deterministic algorithms can do no better than k on any metric space with at least k+1 points, this establishes that for every metric space on which the problem is non-trivial, randomized algorithms give an exponential improvement over deterministic algorithms. Our algorithm maintains an approximation of the underlying metric space by a distribution over HSTs. The granularity and accuracy of the approximation is adjusted dynamically according to the aggregate behavior of the HST algorithms. In short: We try to obtain more accurate approximations at the locations and scales where the "gaction" is happening. Thus a crucial component of our approach is the O((log k)2)-competitive randomized algorithm for HSTs obtained in our previous work with Bubeck, Cohen, Lee, and Ma.dry, and its "multiscale information theory" perspective.

We present an O((logk)2)-competitive randomized algorithm for the k-server problem on hierarchically separated trees (HSTs). This is the first o(k)-competitive randomized algorithm for which the competitive ratio is independent of the size of the underlying HST. Our algorithm is designed in the framework of online mirror descent where the mirror map is a multiscale entropy. When combined with Bartal’s static HST embedding reduction, this leads to an O((logk)2 logn)-competitive algorithm on any n-point metric space. We give a new dynamic HST embedding that yields an O((logk)3 logΔ)-competitive algorithm on any metric space where the ratio of the largest to smallest non-zero distance is at most Δ.

For undirected graphs G=(V,E) and G_0=(V_0,E_0), say that G is a region intersection graph over G_0 if there is a family of connected subsets {R_u \subseteq V_0 : u \in V} of G_0 such that {u,v} \in E \iff R_u \cap R_v \neq \emptyset. We show if G_0 excludes the complete graph K_h as a minor for some h \geq 1, then every region intersection graph G over G_0 with m edges has a balanced separator with at most c_h \sqrt{m} nodes, where c_h is a constant depending only on h. If G additionally has uniformly bounded vertex degrees, then such a separator is found by spectral partitioning. A string graph is the intersection graph of continuous arcs in the plane. String graphs are precisely region intersection graphs over planar graphs. Thus the preceding result implies that every string graph with m edges has a balanced separator of size O(\sqrt{m}). This bound is optimal, as it generalizes the planar separator theorem. It confirms a conjecture of Fox and Pach (2010), and improves over the O(\sqrt{m} \log m) bound of Matousek (2013).

Chang's lemma is a widely employed result in additive combinatorics. It gives bounds on the dimension of the large spectrum of probability distributions on finite abelian groups. Recently, Bloom (2016) presented a powerful variant of Chang's lemma that yields the strongest known quantitative version of Roth's theorem on 3-term arithmetic progressions in dense subsets of the integers. In this note, we show how such theorems can be derived from the approximation of probability measures via entropy maximization.

Smoothing properties of the noise operator on the discrete cube and on Gaussian space have played a pivotal role in many fields. In particular, these smoothing effects have seena broad range of applications in theoretical computer science. We exhibit new regularization properties of the noise operator on Gaussian space. More specifically, we show that the mass on level sets of a probability density decays uniformly under the Ornstein-Uhlenbeck semi group. This confirms positively the Gaussian case of Talagrand's convolution conjecture (1989)on the discrete cube. A major theme is our use of an It o process (the "F"ollmer drift")which can be seen as an entropy-optimal coupling between the Gaussian measure and another given measure on Gaussian space. To analyze this process, we employ stochastic calculus and Girsanov's change of measure formula. The ideas and tools employed here provide a new perspective on hyper contractivity in Gaussian space and the discrete cube. In particular, our work gives a new way of studying "small" sets in product spaces (e.g., Sets of size 2o(n) in the discrete cube) using a form of regularized online gradient descent.

We introduce a method for proving lower bounds on the efficacy of semidefinite programming (SDP) relaxations for combinatorial problems. In particular, we show that the cut, TSP, and stable set polytopes on n-vertex graphs are not the linear image of the feasible region of any SDP (i.e., any spectrahedron) of dimension less than 2nδ, for some constant δ > 0. This result yields the first super-polynomial lower bounds on the semidefinite extension complexity of any explicit family of polytopes.

We study the computational power of general symmetric relaxations for combinatorial optimization problems, both in the linear programming (LP) and semidefinite programming (SDP) case. We show new connections to explicit LP and SDP relaxations, like those obtained from standard hierarchies. Concretely, for kkn) achieve best-possible k approximation guarantees for Max CSPs among all symmetric SDP relaxations of size at most (kn). This result gives the first k lower bounds for symmetric SDPrelaxations of Max CSPs, and indicates that the sum-of-squares method provides the “right” SDP relaxation for this class of problems. Moreover, for k2k) for the traveling salesman problem that achieve per instance best-possible approximation (kn).

Recall the classical hypothesis testing setting with two convex sets of probability distributions P and Q. One receives either n i.i.d. samples from a distribution p ∈ P or from a distribution q ∈ Q and wants to decide from which set the points were sampled. It is known that the optimal exponential rate at which errors decrease can be achieved by a simple maximum-likelihood ratio test which does not depend on p or q, but only on the sets P and Q.

A basic fact in spectral graph theory is that the number of connected components in an undirected graph is equal to the multiplicity of the eigenvalue zero in the Laplacian matrix of the graph. In particular, the graph is disconnected if and only if there are at least two eigenvalues equal to zero. Cheeger's inequality and its variants provide an approximate version of the latter fact; they state that a graph has a sparse cut if and only if there are at least two eigenvalues that are close to zero. It has been conjectured that an analogous characterization holds for higher multiplicities: There are k eigenvalues close to zero if and only if the vertex set can be partitioned into k subsets, each defining a sparse cut. We resolve this conjecture positively. Our result provides a theoretical justification for clustering algorithms that use the bottom k eigenvectors to embed the vertices into R k , and then apply geometric considerations to the embedding. We also show that these techniques yield a nearly optimal quantitative connection between the expansion of sets of size ≈ n / k and λ k , the k th smallest eigenvalue of the normalized Laplacian, where n is the number of vertices. In particular, we show that in every graph there are at least k /2 disjoint sets (one of which will have size at most 2 n / k ), each having expansion at most O (√λ k log k ). Louis, Raghavendra, Tetali, and Vempala have independently proved a slightly weaker version of this last result. The √log k bound is tight, up to constant factors, for the “noisy hypercube” graphs.

The 2-sum embedding conjecture (Lee-Sidiropoulos, FOCS 2009) states that all the shortest-path metrics supported on a family of graphs F admit a uniformly bi-Lipschitz embedding into L1 if and only if the same property holds for the closure of F under edge sums. The problem has an equivalent formulation in terms of multi-commodity flow/cut gaps and appears to be a fundamental step in resolving the well-studied question of which families of graphs admit an approximate multi-commodity max-flow/min-cut theorem.

We prove super-polynomial lower bounds on the size of linear programming relaxations for approximation versions of constraint satisfaction problems. We show that for these problems, polynomial-sized linear programs are exactly as powerful as programs arising from a constant number of rounds of the Sherali-Adams hierarchy. In particular, any polynomial-sized linear program for MAX CUT has an integrality gap of 1/2 and any such linear program for MAX 3-SAT has an integrality gap of 7/8.

The classical Okamura-Seymour theorem states that for an edge-capacitated, multi-commodity flow instance in which all terminals lie on a single face of a planar graph, there exists a feasible concurrent flow if and only if the cut conditions are satisfied. Simple examples show that a similar theorem is impossible in the node-capacitated setting. Nevertheless, we prove that an approximate flow/cut theorem does hold: For some universal ε > 0, if the node cut conditions are satisfied, then one can simultaneously route an ε-fraction of all the demands. This answers an open question of Chekuri and Kawarabayashi. More generally, we show that this holds in the setting of multi-commodity polymatroid networks introduced by Chekuri, et. al. Our approach employs a new type of random metric embedding in order to round the convex programs corresponding to these more general flow problems.

We that every n-point tree metric admits a (1 + ε)-embedding into ℓ1C(ε)log n, for every ε > 0, where C(ε) ≤ . This matches the natural volume lower bound up to a factor depending only on ε. Previously it was unknown whether even complete binary trees on n nodes could be embedded in ℓ1O(log n) with O(1) distortion. For complete d-ary trees, our construction achieves .

A basic fact in spectral graph theory is that the number of connected components in an undirected graph is equal to the multiplicity of the eigenvalue zero in the Laplacian matrix of the graph. In particular, the graph is disconnected if and only if there are at least two eigenvalues equal to zero. Cheeger's inequality and its variants provide an approximate version of the latter fact; they state that a graph has a sparse cut if and only if there are at least two eigenvalues that are close to zero.

We exhibit a strong connection between cover times of graphs, Gaussian processes, and Talagrand's theory of majorizing measures. In particular, we show that the cover time of any graph G is equivalent, up to universal constants, to the square of the expected maximum of the Gaussian free field on G, scaled by the number of edges in G.

We exhibit an infinite doubling metric space (X,d) such that for any non-expansive f : X -> L1, there exists a pair x,y ∈ X with d(x,y) arbitrarily large, and such that |f(x)-f(y)\|1/d(x,y) ≲ √log log d(x,y)}/(log d(x,y)).

We study the quantitative geometry of graphs in terms of their genus, using the structure of certain “cut graphs,” i.e. subgraphs whose removal leaves a planar graph. In particular, we give optimal bounds for random partitioning schemes, as well as various types of embeddings. Using these geometric primitives, we present exponentially improved dependence on genus for a number of problems like approximate max-flow/min-cut theorems, approximations for uniform and non-uniform Sparsest Cut, treewidth approximation, Laplacian eigenvalue bounds, and Lipschitz extension theorems and related metric labeling problems. We list here a sample of these improvements. All the following statements refer to graphs of genus g, unless otherwise noted. We show that such graphs admit an O(log g)-approximate multi-commodity max-flow/min-cut theorem for the case of uniform demands. This bound is optimal, and improves over the previous bound of O(g) [KPR93, FT03]. For general demands, we show that the worst possible gap is O(log g + CP), where CP is the gap for planar graphs. This dependence is optimal, and already yields a bound of , improving over the previous bound of [KLMN04]. We give an -approximation for the uniform Sparsest Cut, balanced vertex separator, and treewidth problems, improving over the previous bound of O(g) [FHL05]. If a graph G has genus g and maximum degree D, we show that the kth Laplacian eigenvalue of G is (log g)2 · O(kg D/n), improving over the previous bound of g2 · O(kg D/n) [KLPT09]. There is a lower bound of Ω(kg D/n), making this result almost tight. We show that if (X, d) is the shortest-path metric on a graph of genus g and S ⊆ X, then every L-Lipschitz map f: S → Z into a Banach space Z admits an O(L log g)-Lipschitz extension . This improves over the previous bound of O(Lg) [LN05], and compares to a lower bound of . In a related way, we show that there is an O(log g)-approximation for the 0-extension problem on such graphs, improving over the previous O(g) bound. We show that every n-vertex shortest-path metric on a graph of genus g embeds into L2 with distortion , improving over the previous bound of . Our result is asymptotically optimal for every dependence g = g(n).

We show that there exists a metric space (X,d) such that (X,√d) admits a bilipschitz embedding into L2, but (X,d) does not admit an equivalent metric of negative type. In fact, we exhibit a strong quantitative bound: There are n-point subsets Yn ⊆ X such that mapping (Yn, d) to a metric of negative type requires distortion ~Ω(log n)1/4.

We present a new method for upper bounding the second eigenvalue of the Laplacian of graphs. Our approach uses multi-commodity flows to deform the geometry of the graph; we embed the resulting metric into Euclidean space to recover a bound on the Rayleigh quotient. Using this, we show that every n -vertex graph of genus g and maximum degree D satisfies λ 2 ( G )= O (( g +1) 3 D / n ). This recovers the O ( D / n ) bound of Spielman and Teng for planar graphs, and compares to Kelner's bound of O (( g +1)poly( D )/ n ), but our proof does not make use of conformal mappings or circle packings. We are thus able to extend this to resolve positively a conjecture of Spielman and Teng, by proving that λ 2 ( G ) = O ( D h 6 log h / n ) whenever G is K h -minor free. This shows, in particular, that spectral partitioning can be used to recover O (√ n )-sized separators in bounded degree graphs that exclude a fixed minor. We extend this further by obtaining nearly optimal bounds on λ 2 for graphs that exclude small-depth minors in the sense of Plotkin, Rao, and Smith. Consequently, we show that spectral algorithms find separators of sublinear size in a general class of geometric graphs. Moreover, while the standard “sweep” algorithm applied to the second eigenvector may fail to find good quotient cuts in graphs of unbounded degree, our approach produces a vector that works for arbitrary graphs. This yields an alternate proof of the well-known nonplanar separator theorem of Alon, Seymour, and Thomas that states that every excluded-minor family of graphs has O (√ n )-node balanced separators.

This special section comprises eight fully refereed papers whose extended abstracts were presented at the 48th Annual IEEE Symposium on Foundations of Computer Science (FOCS 2007) in Providence, Rhode Island, October 21–23, 2007. The unrefereed conference versions of these papers were published by IEEE in the FOCS 2007 proceedings. The regular conference program consisted of 63 papers chosen from among 302 submissions. These were selected by a program committee consisting of Dimitris Achlioptas, Timothy Chan, Julia Chuzhoy, Faith Ellen, Piotr Indyk, Kamal Jain, T. S. Jayram, Robert Kleinberg, James R. Lee, Anna Lysyanskaya, Daniele Micciancio, Gary Miller, Moni Naor, Alexander Razborov, Yaoyun Shi, Alistair Sinclair (chair), Luca Trevisan, Chris Umans, and Uri Zwick. The papers invited to this special section were also selected with the input of the program committee. The eight papers in this section span a broad range of topics, including algorithmic game theory, communication complexity, hardness of approximation, metric embeddings, proof complexity, pseudorandomness, and quantum algorithms. Each paper underwent an extensive refereeing process; we thank both the authors and the anonymous referees for their efforts. In addition, we would like to thank Eva Tardos, who was SICOMP's editor-in-chief during the course of this project, and SIAM staff members Mitch Chernoff and Cherie Trebisky for their help in preparing this special section.

It was shown in [Indyk-Sidiropoulos 07] that any orientable graph of genus g can be probabilistically embedded into a graph of genus g-1 with constant distortion. Removing handles one by one gives an embedding into a distribution over planar graphs with distortion 2O(g). By removing all $g$ handles at once, we present a probabilistic embedding with distortion O(g2) for both orientable and non-orientable graphs. Our result is obtained by showing that the minimum-cut graph of [Erickson-HarPeled 04] has low dilation, and then randomly cutting this graph out of the surface using the Peeling Lemma from [Lee-Sidiropoulos 08].

We present a general method for proving upper bounds on the eigenvalues of the graph Laplacian. In particular, we show that for any positive integer k, the kthsmallest eigenvalue of the Laplacian on a bounded-degree planar graph is O(k/n). This bound is asymptotically tight for every k, as it is easily seen to be achieved for planar grids. We also extend this spectral result to graphs with bounded genus, graphs which forbid fixed minors, and other natural families. Previously, such spectral upper bounds were only known for k = 2, i.e. for the Fiedler value of these graphs. In addition, our result yields a new, combinatorial proof of the celebrated result of Korevaar in differential geometry.

We study the topological simplification of graphs via random embeddings, leading ultimately to a reduction of the Gupta-Newman-Rabinovich-Sinclair (GNRS) L1 embedding conjecture to a pair of manifestly simpler conjectures. The GNRS conjecture characterizes all graphs that have an O(1)-approximate multi-commodity max-flow/min-cut theorem. In particular, its resolution would imply a constant factor approximation for the general Sparsest Cut problem in every family of graphs which forbids some minor. In the course of our study, we prove a number of results of independent interest.

We present a new method for upper bounding the second eigenvalue of theLaplacian of graphs. Our approach uses multi-commodity flows to deform the geometry of the graph; we embed the resulting metric into Euclidean space to recover a bound on the Rayleigh quotient. Using this, we show that every n-vertex graph of genus g and maximum degree d satisfies lambda2(G) = O((g+1)3d/n).This recovers the O(d/n) bound of Spielman and Teng for planar graphs, and compares to Kelner's bound of O((g+1)poly(d)/n), but our proof does not make use of conformal mappings or circle packings. We are thus able to extend this to resolve positively a conjecture of Spielman and Teng, by proving that lambda2(G) = O(dh6log h/n) whenever G is Kh-minor free. This shows, in particular, that spectral partitioning can be used to recover O(radicn)-sized separators in bounded degree graphs that exclude a fixed minor. We extend this further by obtaining nearly optimal bounds on lambda2for graphs which exclude small-depth minors in the sense of Plotkin, Rao, and Smith. Consequently, we show that spectral algorithms find small separators in a general class of geometric graphs. Moreover, while the standard "sweep'' algorithm applied to the second eigenvector may fail to find good quotient cuts in graphs of unbounded degree, our approach produces a vector that works for arbitrary graphs. This yields an alternate proof of the result of Alon, Seymour, and Thomas that every excluded-minor family of graphs has O(radicn)-node balanced separators.

We study the properties of embeddings, multicommodity flows, and sparse cuts in minor-closed families of graphs which are also closed under 2-sums; this includes planar graphs, graphs of bounded treewidth, and constructions based on recursive edge replacement.

We develop the algorithmic theory of vertex separators and its relation to the embeddings of certain metric spaces. Unlike in the edge case, we show that embeddings into $L_1$ (and even Euclidean embeddings) are insufficient but that the additional structure provided by many embedding theorems does suffice for our purposes. We obtain an $O(\sqrt{\log n})$ approximation for minimum ratio vertex cuts in general graphs, based on a new semidefinite relaxation of the problem, and a tight analysis of the integrality gap which is shown to be $\Theta(\sqrt{\log n})$. We also prove an optimal $O(\log k)$-approximate max-flow/min-vertex-cut theorem for arbitrary vertex-capacitated multicommodity flow instances on k terminals. For uniform instances on any excluded-minor family of graphs, we improve this to $O(1)$, and this yields a constant-factor approximation for minimum ratio vertex cuts in such graphs. Previously, this was known only for planar graphs, and for general excluded-minor families the best known ratio was $O(\log n)$. These results have a number of applications. We exhibit an $O(\sqrt{\log n})$ pseudoapproximation for finding balanced vertex separators in general graphs. In fact, we achieve an approximation ratio of $O(\sqrt{\log {opt}})$, where ${opt}$ is the size of an optimal separator, improving over the previous best bound of $O(\log {opt})$. Likewise, we obtain improved approximation ratios for treewidth: in any graph of treewidth k, we show how to find a tree decomposition of width at most $O(k \sqrt{\log k})$, whereas previous algorithms yielded $O(k \log k)$. For graphs excluding a fixed graph as a minor (which includes, e.g., bounded genus graphs), we give a constant-factor approximation for the treewidth. This in turn can be used to obtain polynomial-time approximation schemes for several problems in such graphs.

We consider questions about vertex cuts in graphs, random walks in metric spaces, and dimension reduction in L1 and L2; these topics are intimately connected because they can each be reduced to the existence ofvarious families of real-valued Lipschitz maps on certain metric spaces. We view these issues through the lens of shortest-path metricson series-parallel graphs, and we discussthe implications for a variety of well-known open problems. Our main results follow.

In [Rao 1999], it is shown that every n-point Euclidean metric with polynomial aspect ratio admits a Euclidean embedding with k-dimensional distortion bounded by O ( √ log n log k), a result which is tight for constant values of k. We show that this holds without any assumption on the aspect ratio, and give an improved bound of O ( √ log n(log k) 1/4). Our main result is an upper bound of O ( √ log n log log n) independent of the value of k, nearly resolving the main open questions of [Dunagan-Vempala 2001] and [Krauthgamer-Linial-Magen 2004]. The best previous bound was O(log n), and our bound is nearly tight, as even the 2-dimensional volume distortion of an n-vertex path is Ω ( √ log n). 1

We initiate the study of approximate algorithms on negatively curved spaces. These spaces have recently become of interest in various domains of computer science including networking and vision. The classical example of such a space is the real-hyperbolic space \mathbb{H}^d for d \geqslant 2, but our approach applies to a more general family of spaces characterized by Gromov's (combinatorial) hyperbolic condition. We give efficient algorithms and data structures for problems like approximate nearest-neighbor search and compact, low-stretch routing on subsets of negatively curved spaces of fixed dimension (including \mathbb{H}^d as a special case). In a different direction, we show that there is a PTAS for the Traveling Salesman Problem when the set of cities lie, for example, in \mathbb{H}^d. This generalizes Arora's results for \mathbb{R}^d. Most of our algorithms use the intrinsic distance geometry of the data set, and only need the existence of an embedding into some negatively curved space in order to function properly. In other words, our algorithms regard the interpoint distance function as a black box, and are independent of the representation of the input points.

We prove that the function d : Ropf3times Ropf3rarr [0,infin] given by d((x,y,z),(t,u,v)) = ([((t-x)2+(u-y)2)2+ (v-z+2xu-2yt)2]frac12+ (t-x)2+ (u-y)2)frac12is a metric on Ropf3such that (Ropf3,radicd) is isometric to a subset of Hilbert space, yet (Ropf3, d) does not admit a bi-Lipschitz embedding into L1. This yields a new simple counter example to the Goemans-Linial conjecture on the integrality gap of the semidefinite relaxation of the sparsest cut problem. The metric above is doubling, and hence has a padded stochastic decomposition at every scale. We also study the Lpversion of this problem, and obtain a counter example to a natural generalization of a classical theorem of Bretagnolle et al. (1996) (of which the Goemans-Linial conjecture is a particular case). Our methods involve Fourier analytic techniques, and a breakthrough of Cheeger and Kleiner (2006), together with classical results of Pansu (1989) on the differentiability of Lipschitz functions on the Heisenberg group

We prove that every n-point metric space of negative type (in particular, every n-point subset of L1) embeds into a Euclidean space with distortion O(√log n log log n), a result which is tight up to the O(log log n) factor. As a consequence, we obtain the best known polynomial-time approximation algorithm for the Sparsest Cut problem with general demands. If the demand is supported on a subset of size k, we achieve an approximation ratio of O(√log k log log k).

We develop the algorithmic theory of vertex separators, and its relation to the embeddings of certain metric spaces. Unlike in the edge case, we show that embeddings into L1 (and even Euclidean embeddings) are insufficient, but that the additional structure provided by many embedding theorems does suffice for our purposes.We obtain an O(√log n) approximation for min-ratio vertex cuts in general graphs, based on a new semidefinite relaxation of the problem, and a tight analysis of the integrality gap which is shown to be Θ(√log n). We also prove various approximate max-flow/min-vertex-cut theorems, which in particular give a constant-factor approximation for min-ratio vertex cuts in any excluded-minor family of graphs. Previously, this was known only for planar graphs, and for general excluded-minor families the best-known ratio was O(log n).These results have a number of applications. We exhibit an O(√log n) pseudo-approximation for finding balanced vertex separators in general graphs. In fact, we achieve an approximation ratio of O(√log opt) where opt is the size of an optimal separator, improving over the previous best bound of O(log opt). Likewise, we obtain improved approximation ratios for treewidth: In any graph of treewidth k, we show how to find a tree decomposition of width at most O(k √log k), whereas previous algorithms yielded O(k log k). For graphs excluding a fixed graph as a minor (which includes, e.g., bounded genus graphs), we give a constant-factor approximation for the treewidth; this can be used to obtain the first polynomial-time approximation schemes for problems like minimum feedback vertex set and minimum connected dominating set in such graphs.

We devise a new embedding technique, which we call measured descent, based on decomposing a metric space locally, at varying speeds, according to the density of some probability measure. This provides a refined and unified framework for the two primary methods of constructing Frechet embeddings for finite metrics, due to J. Bourgain and S. Rao. We prove that any n-point metric space (X, d) embeds in Hilbert space with distortion O(/spl radic//spl alpha//sub X//spl middot/log n), where /spl alpha//sub X/ is a geometric estimate on the decomposability of X. An an immediate corollary, we obtain an O(/spl radic/log /spl lambda//sub X//spl middot/log n) distortion embedding, where /spl lambda//sub X/ is the doubling constant of X. Since /spl lambda//sub X/ /spl les/ n, this result recovers Bourgain 5 theorem, but when the metric X is, in a sense, "low-dimensional", improved bounds are achieved. Our embeddings are volume-respecting for subsets of arbitrary size. One consequence is the existence of (k, O(log n)) volume-respecting embeddings for all 1 /spl les/ k /spl les/ n, which is the best possible, and answers positively a question posed by U. Feige. Our techniques are also used to answer positively a question of Y. Rabinovich, showing that any weighted n-point planar graph embeds in /spl lscr//sub /spl infin///sup O(log n)/ with O(1) distortion. The O(log n) bound on the dimension is optimal, and improves upon the previously known bound of O(log/sup 2/ n).

In the survivable networkdesign problem (SNDP), the goal is to find a minimum-cost spanning subgraph satisfying certain connectivity requirements. We study the vertex-connectivity variant of SNDP in which the input specifies, for each pair of vertices, a required number of vertex-disjoint paths connecting them. We give the first strong lower bound on the approximability of SNDP, showing that the problem admits no efficient $2^{\log^{1-\epsilon} n}$ ratio approximation for any fixed $\epsilon\! >\! 0$, unless $\NP\subseteq \DTIME(n^{\polylog(n)})$. We show hardness of approximation results for some important special cases of SNDP, and we exhibit the first lower bound on the approximability of the related classical NP-hard problem of augmenting the connectivity of a graph using edges from a given set.

The doubling constant of a metric space (X, d) is the smallest value /spl lambda/ such that every ball in X can be covered by /spl lambda/ balls of half the radius. The doubling dimension of X is then defined as dim (X) = log/sub 2//spl lambda/. A metric (or sequence of metrics) is called doubling precisely when its doubling dimension is bounded. This is a robust class of metric spaces which contains many families of metrics that occur in applied settings. We give tight bounds for embedding doubling metrics into (low-dimensional) normed spaces. We consider both general doubling metrics, as well as more restricted families such as those arising from trees, from graphs excluding a fixed minor, and from snowflaked metrics. Our techniques include decomposition theorems for doubling metrics, and an analysis of a fractal in the plane according to T. J. Laakso (2002). Finally, we discuss some applications and point out a central open question regarding dimensionality reduction in L/sub 2/.

We resolve the following conjecture raised by Levin together with Linial, London, and Rabinovich [16]. Let Z∞d be the infinite graph whose vertex set is Zd and which has an edge (u,v) whenever ||u-v||∞ = 1. Let dim(G) be the smallest d such that G occurs as a (not necessarily induced) subgraph of Z∞d. The growth rate of G, denoted ρG, is the minimum ρ such that every ball of radius r > 1 in G contains at most rρ vertices. By simple volume arguments, dim(G) = Ω(ρG). Levin conjectured that this lower bound is tight, i.e., that dim(G) = O(ρG) for every graph G.Previously, it was not known whether dim(G) could be upper bounded by any function of ρG, even in the special case of trees. We show that a weaker form of Levin's conjecture holds by proving that, for every graph G, dim(G) = O(ρG log ρG). We disprove, however, the specific bound of the conjecture and show that our upper bound is tight by exhibiting graphs for which dim(G) =Ω(ρG log ρG). For families of graphs which exclude a fixed minor, we salvage the strong form, showing that dim(G) = O(ρG). This holds also for graphs without long induced simple cycles. Our results extend to a variant of the conjecture for finite-dimensional Euclidean spaces due to Linial[15].