Results 21  30
of
312
Embeddings of negativetype metrics and an improved approximation to generalized sparsest cut
, 2007
"... In this paper, we study metrics of negative type, which are metrics (V, d) such that √ d is an Euclidean metric; these metrics are thus also known as “ℓ2squared” metrics. We show how to embed npoint negativetype metrics into Euclidean space ℓ2 with distortion D = O(log 3/4 n). This embedding resu ..."
Abstract

Cited by 49 (0 self)
 Add to MetaCart
(Show Context)
In this paper, we study metrics of negative type, which are metrics (V, d) such that √ d is an Euclidean metric; these metrics are thus also known as “ℓ2squared” metrics. We show how to embed npoint negativetype metrics into Euclidean space ℓ2 with distortion D = O(log 3/4 n). This embedding result, in turn, implies an O(log 3/4 k)approximation algorithm for the Sparsest Cut problem with nonuniform demands. Another corollary we obtain is that npoint subsets of ℓ1 embed into ℓ2 with distortion O(log 3/4 n).
The bidimensionality Theory and Its Algorithmic Applications
 Computer Journal
, 2005
"... This paper surveys the theory of bidimensionality. This theory characterizes a broad range of graph problems (‘bidimensional’) that admit efficient approximate or fixedparameter solutions in a broad range of graphs. These graph classes include planar graphs, map graphs, boundedgenus graphs and gra ..."
Abstract

Cited by 47 (3 self)
 Add to MetaCart
This paper surveys the theory of bidimensionality. This theory characterizes a broad range of graph problems (‘bidimensional’) that admit efficient approximate or fixedparameter solutions in a broad range of graphs. These graph classes include planar graphs, map graphs, boundedgenus graphs and graphs excluding any fixed minor. In particular, bidimensionality theory builds on the Graph Minor Theory of Robertson and Seymour by extending the mathematical results and building new algorithmic tools. Here, we summarize the known combinatorial and algorithmic results of bidimensionality theory with the highlevel ideas involved in their proof; we describe the previous work on which the theory is based and/or extends; and we mention several remaining open problems. 1.
Finding sparse cuts locally using evolving sets
 In STOC'09: Proceedings of the 41st Annual ACM symposium on Theory of Computing
, 2009
"... A local graph partitioning algorithm finds a set of vertices with small conductance (i.e. a sparse cut) by adaptively exploring part of a large graph G, starting from a specified vertex. For the algorithm to be local, its complexity must be bounded in terms of the size of the set that it outputs, wi ..."
Abstract

Cited by 42 (0 self)
 Add to MetaCart
(Show Context)
A local graph partitioning algorithm finds a set of vertices with small conductance (i.e. a sparse cut) by adaptively exploring part of a large graph G, starting from a specified vertex. For the algorithm to be local, its complexity must be bounded in terms of the size of the set that it outputs, with at most a weak dependence on the number n of vertices in G. Previous local partitioning algorithms find sparse cuts using random walks and personalized PageRank. In this paper, we introduce a randomized local partitioning algorithm that finds a sparse cut by simulating the volumebiased evolving set process, which is a Markov chain on sets of vertices. We prove that for any set of vertices A that has conductance at most φ, for at least half of the starting vertices in A our algorithm will output (with probability at least half), a set of conductance O(φ 1/2 log 1/2 n). We prove that for a given run of the algorithm, the expected ratio between its computational complexity and the volume of the set that it outputs is O(φ −1/2 polylog(n)). In comparison, the best previous local partitioning algorithm, due to Andersen, Chung, and Lang, has the same approximation guarantee, but a larger ratio of O(φ −1 polylog(n)) between the complexity and output volume. Using our local partitioning algorithm as a subroutine, we construct a fast algorithm for finding balanced cuts. Given a fixed value of φ, the resulting algorithm has complexity (m + nφ −1/2)) · O(polylog(n)) and returns a cut with conductance O(φ 1/2 log 1/2 n) and volume at least vφ/2, where vφ is the largest volume of any set with conductance at most φ. 1 1
Fast Algorithms for Approximate Semidefinite Programming using the Multiplicative Weights Update Method
"... Semidefinite programming (SDP) relaxations appear inmany recent approximation algorithms but the only general technique for solving such SDP relaxations is via interior point methods. We use a Lagrangianrelaxation based technique (modified from the papers of Plotkin, Shmoys,and Tardos (PST), and ..."
Abstract

Cited by 41 (6 self)
 Add to MetaCart
(Show Context)
Semidefinite programming (SDP) relaxations appear inmany recent approximation algorithms but the only general technique for solving such SDP relaxations is via interior point methods. We use a Lagrangianrelaxation based technique (modified from the papers of Plotkin, Shmoys,and Tardos (PST), and Klein and Lu) to derive faster algorithms for approximately solving several families of SDPrelaxations. The algorithms are based upon some improvements to the PST ideas which lead to new results even fortheir framework as well as improvements in approximate eigenvalue computations by using random sampling.
Graph Expansion and the Unique Games Conjecture
, 2010
"... The edge expansion of a subset of vertices S ⊆ V in a graph G measures the fraction of edges that leave S. In a dregular graph, the edge expansion/conductance Φ(S) of a subset E(S,V \S) dS S ⊆ V is defined as Φ(S) =. Approximating the conductance of small linear sized sets (size δn) is a natur ..."
Abstract

Cited by 41 (5 self)
 Add to MetaCart
The edge expansion of a subset of vertices S ⊆ V in a graph G measures the fraction of edges that leave S. In a dregular graph, the edge expansion/conductance Φ(S) of a subset E(S,V \S) dS S ⊆ V is defined as Φ(S) =. Approximating the conductance of small linear sized sets (size δn) is a natural optimization question that is a variant of the wellstudied Sparsest Cut problem. However, there are no known algorithms to even distinguish between almost complete edge expansion (Φ(S) = 1 − ε), and close to 0 expansion. In this work, we investigate the connection between Graph Expansion and the Unique Games Conjecture. Specifically, we show the following: –We show that a simple decision version of the problem of approximating small set expansion reduces to Unique Games. Thus if approximating edge expansion of small sets is hard, then Unique Games is hard. Alternatively, a refutation of the UGC will yield better algorithms to approximate edge expansion in graphs. This is the first nontrivial “reverse ” reduction from a natural optimization problem to Unique Games. –Under a slightly stronger UGC that assumes mild expansion of small sets, we show that it is UGhard to approximate small set expansion. –On instances with sufficiently good expansion of small sets, we show that Unique Games is easy by extending the techniques of [4].
Submodular Approximation: Samplingbased Algorithms and Lower Bounds
, 2008
"... We introduce several generalizations of classical computer science problems obtained by replacing simpler objective functions with general submodular functions. The new problems include submodular load balancing, which generalizes load balancing or minimummakespan scheduling, submodular sparsest cu ..."
Abstract

Cited by 40 (0 self)
 Add to MetaCart
(Show Context)
We introduce several generalizations of classical computer science problems obtained by replacing simpler objective functions with general submodular functions. The new problems include submodular load balancing, which generalizes load balancing or minimummakespan scheduling, submodular sparsest cut and submodular balanced cut, which generalize their respective graph cut problems, as well as submodular function minimization with a cardinality lower bound. We establish upper and lower bounds for the approximability of these problems with a polynomial number of queries to a functionvalue oracle. The approximation guarantees for most of our algorithms are of the order of √ n/lnn. We show that this is the inherent difficulty of the problems by proving matching lower bounds. We also give an improved lower bound for the problem of approximately learning a monotone submodular function. In addition, we present an algorithm for approximately learning submodular functions with special structure, whose guarantee is close to the lower bound. Although quite restrictive, the class of functions with this structure includes the ones that are used for lower bounds both by us and in previous work. This demonstrates that if there are significantly stronger lower bounds for this problem, they rely on more general submodular functions.
Improved lower bounds for embeddings into L1
 SIAM J. COMPUT.
, 2009
"... We improve upon recent lower bounds on the minimum distortion of embedding certain finite metric spaces into L1. In particular, we show that for every n ≥ 1, there is an npoint metric space of negative type that requires a distortion of Ω(log log n) for such an embedding, implying the same lower bo ..."
Abstract

Cited by 39 (5 self)
 Add to MetaCart
(Show Context)
We improve upon recent lower bounds on the minimum distortion of embedding certain finite metric spaces into L1. In particular, we show that for every n ≥ 1, there is an npoint metric space of negative type that requires a distortion of Ω(log log n) for such an embedding, implying the same lower bound on the integrality gap of a wellknown semidefinite programming relaxation for sparsest cut. This result builds upon and improves the recent lower bound of (log log n) 1/6−o(1) due to Khot and Vishnoi [The unique games conjecture, integrality gap for cut problems and the embeddability of negative type metrics into l1, in Proceedings of the 46th Annual IEEE Symposium
Energy Models for Graph Clustering
"... The cluster structure of many realworld graphs is of great interest, as the clusters may correspond e.g. to communities in social networks or to cohesive modules in software systems. Layouts can naturally represent the cluster structure of graphs by grouping densely connected nodes and separating s ..."
Abstract

Cited by 38 (1 self)
 Add to MetaCart
(Show Context)
The cluster structure of many realworld graphs is of great interest, as the clusters may correspond e.g. to communities in social networks or to cohesive modules in software systems. Layouts can naturally represent the cluster structure of graphs by grouping densely connected nodes and separating sparsely connected nodes. This article introduces two energy models whose minimum energy layouts represent the cluster structure, one based on repulsion between nodes (like most existing energy models) and one based on repulsion between edges. The latter model is not biased towards grouping nodes with high degrees, and is thus more appropriate for the many realworld graphs with rightskewed degree distributions. The two energy models are shown to be closely related to widely used quality criteria for graph clusterings – namely the density of the cut, Shi and Malik’s normalized cut, and Newman’s modularity – and to objective functions optimized by eigenvectorbased graph drawing methods.
Advances in metric embedding theory
 IN STOC ’06: PROCEEDINGS OF THE THIRTYEIGHTH ANNUAL ACM SYMPOSIUM ON THEORY OF COMPUTING
, 2006
"... Metric Embedding plays an important role in a vast range of application areas such as computer vision, computational biology, machine learning, networking, statistics, and mathematical psychology, to name a few. The theory of metric embedding received much attention in recent years by mathematicians ..."
Abstract

Cited by 36 (13 self)
 Add to MetaCart
Metric Embedding plays an important role in a vast range of application areas such as computer vision, computational biology, machine learning, networking, statistics, and mathematical psychology, to name a few. The theory of metric embedding received much attention in recent years by mathematicians as well as computer scientists and has been applied in many algorithmic applications. A cornerstone of the field is a celebrated theorem of Bourgain which states that every finite metric space on n points embeds in Euclidean space with O(log n) distortion. Bourgain’s result is best possible when considering the worst case distortion over all pairs of points in the metric space. Yet, it is possible that an embedding can do much better in terms of the average distortion. Indeed, in most practical applications of metric embedding the main criteria for the quality of an embedding is its average distortion over all pairs. In this paper we provide an embedding with constant average distortion for arbitrary metric spaces, while maintaining the same worst case bound provided by Bourgain’s theorem. In fact, our embedding possesses a much stronger property. We define the ℓqdistortion of a uniformly distributed pair of points. Our embedding achieves the best possible ℓqdistortion for all 1 ≤ q ≤ ∞ simultaneously. These results have several algorithmic implications, e.g. an O(1) approximation for the unweighted uncapacitated quadratic assignment problem. The results are based on novel embedding methods which improve on previous methods in another important aspect: the dimension. The dimension of an embedding is of very high importance in particular in applications and much effort has been invested in analyzing it. However, no previous result im