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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 ..."
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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.
On partitioning graphs via single commodity flows
 In STOC ’08: Proceedings of the 40th Annual ACM Symposium on Theory of Computing
, 2008
"... In this paper we obtain improved upper and lower bounds for the best approximation factor for Sparsest Cut achievable in the cutmatching game framework proposed in Khandekar et al. [9]. We show that this simple framework can be used to design combinatorial algorithms that achieve O(log n) approxima ..."
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Cited by 36 (6 self)
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In this paper we obtain improved upper and lower bounds for the best approximation factor for Sparsest Cut achievable in the cutmatching game framework proposed in Khandekar et al. [9]. We show that this simple framework can be used to design combinatorial algorithms that achieve O(log n) approximation factor and whose running time is dominated by a polylogarithmic number of singlecommodity maxflow computations. This matches the performance of the algorithm of Arora and Kale [2]. Moreover, we also show that it is impossible to get an approximation factor of better than Ω ( √ log n) in the cutmatching game framework. These results suggest that the simple and concrete abstraction of the cutmatching game may be powerful enough to capture the essential features of the complexity of Sparsest Cut. Categories and Subject Descriptors
Fixedparameter tractability of multicut parameterized by the size of the cutset
, 2011
"... Given an undirected graph G, a collection {(s1, t1),...,(sk, tk)} of pairs of vertices, and an integer p, the EDGE MULTICUT problem ask if there is a set S of at most p edges such that the removal of S disconnects every si from the corresponding ti. VERTEX MULTICUT is the analogous problem where S i ..."
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Cited by 33 (5 self)
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Given an undirected graph G, a collection {(s1, t1),...,(sk, tk)} of pairs of vertices, and an integer p, the EDGE MULTICUT problem ask if there is a set S of at most p edges such that the removal of S disconnects every si from the corresponding ti. VERTEX MULTICUT is the analogous problem where S is a set of at most p vertices. Our main result is that both problems can be solved in time 2O(p3) · nO(1), i.e., fixedparameter tractable parameterized by the size p of the cutset in the solution. By contrast, it is unlikely that an algorithm with running time of the form f (p) · nO(1) exists for the directed version of the problem, as we show it to be W[1]hard parameterized by the size of the cutset.
New approximation guarantee for chromatic number
 STOC'06
, 2006
"... We describe how to color every 3colorable graph with O(n0.2111) colors, thus improving an algorithm of Blum and Karger from almost a decade ago. Our analysis uses new geometric ideas inspired by the recent work of Arora, Rao, and Vazirani on SPARSEST CUT, and these ideas show promise of leading to ..."
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Cited by 32 (3 self)
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We describe how to color every 3colorable graph with O(n0.2111) colors, thus improving an algorithm of Blum and Karger from almost a decade ago. Our analysis uses new geometric ideas inspired by the recent work of Arora, Rao, and Vazirani on SPARSEST CUT, and these ideas show promise of leading to further improvements.
Breaking the multicommodity flow barrier for O( √ logn)approximations to sparsest cut
 In FOCS
, 2009
"... This paper ties the line of work on algorithms that find an O ( √ log n)approximation to the sparsest cut together with the line of work on algorithms that run in subquadratic time by using only singlecommodity flows. We present an algorithm that simultaneously achieves both goals, finding an O ..."
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Cited by 31 (0 self)
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This paper ties the line of work on algorithms that find an O ( √ log n)approximation to the sparsest cut together with the line of work on algorithms that run in subquadratic time by using only singlecommodity flows. We present an algorithm that simultaneously achieves both goals, finding an O ( √ log(n)/ε)approximation using O(n ε log O(1) n) maxflows. The core of the algorithm is a stronger, algorithmic version of Arora et al.’s structure theorem, where we show that matchingchaining argument at the heart of their proof can be viewed as an algorithm that finds good augmenting paths in certain geometric multicommodity flow networks. By using that specialized algorithm in place of a blackbox solver, we are able to solve those instances much more efficiently. We also show the cutmatching game framework can not achieve an approximation any better than Ω(log(n) / log log(n)) without rerouting flow. 1
Algorithms, Graph Theory, and Linear Equations in Laplacian Matrices
"... Abstract. The Laplacian matrices of graphs are fundamental. In addition to facilitating the application of linear algebra to graph theory, they arise in many practical problems. In this talk we survey recent progress on the design of provably fast algorithms for solving linear equations in the Lapla ..."
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Abstract. The Laplacian matrices of graphs are fundamental. In addition to facilitating the application of linear algebra to graph theory, they arise in many practical problems. In this talk we survey recent progress on the design of provably fast algorithms for solving linear equations in the Laplacian matrices of graphs. These algorithms motivate and rely upon fascinating primitives in graph theory, including lowstretch spanning trees, graph sparsifiers, ultrasparsifiers, and local graph clustering. These are all connected by a definition of what it means for one graph to approximate another. While this definition is dictated by Numerical Linear Algebra, it proves useful and natural from a graph theoretic perspective.
Graph partitioning by spectral rounding: Applications in image segmentation and clustering
 In CVPR
, 2006
"... We introduce a family of spectral partitioning methods. Edge separators of a graph are produced by iteratively reweighting the edges until the graph disconnects into the prescribed number of components. At each iteration a small number of eigenvectors with small eigenvalue are computed and used to d ..."
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Cited by 30 (4 self)
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We introduce a family of spectral partitioning methods. Edge separators of a graph are produced by iteratively reweighting the edges until the graph disconnects into the prescribed number of components. At each iteration a small number of eigenvectors with small eigenvalue are computed and used to determine the reweighting. In this way spectral rounding directly produces discrete solutions where as current spectral algorithms must map the continuous eigenvectors to discrete solutions by employing a heuristic geometric separator (e.g. kmeans). We show that spectral rounding compares favorably to current spectral approximations on the Normalized Cut criterion (NCut). Results are given for natural image segmentation, medical image segmentation, and clustering. A practical version is shown to converge. 1.
Dynamics of Large Networks
, 2008
"... A basic premise behind the study of large networks is that interaction leads to complex collective behavior. In our work we found very interesting and counterintuitive patterns for time evolving networks, which change some of the basic assumptions that were made in the past. We then develop models ..."
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Cited by 30 (0 self)
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A basic premise behind the study of large networks is that interaction leads to complex collective behavior. In our work we found very interesting and counterintuitive patterns for time evolving networks, which change some of the basic assumptions that were made in the past. We then develop models that explain processes which govern the network evolution, fit such models to real networks, and use them to generate realistic graphs or give formal explanations about their properties. In addition, our work has a wide range of applications: it can help us spot anomalous graphs and outliers, forecast future graph structure and run simulations of network evolution. Another important aspect of our research is the study of “local ” patterns and structures of propagation in networks. We aim to identify building blocks of the networks and find the patterns of influence that these blocks have on information or virus propagation over the network. Our recent work included the study of the spread of influence in a large persontoperson
Structure Preserving Embedding
"... Structure Preserving Embedding (SPE) is an algorithm for embedding graphs in Euclidean space such that the embedding is lowdimensional and preserves the global topological properties of the input graph. Topology is preserved if a connectivity algorithm, such as knearest neighbors, can easily recove ..."
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Cited by 29 (6 self)
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Structure Preserving Embedding (SPE) is an algorithm for embedding graphs in Euclidean space such that the embedding is lowdimensional and preserves the global topological properties of the input graph. Topology is preserved if a connectivity algorithm, such as knearest neighbors, can easily recover the edges of the input graph from only the coordinates of the nodes after embedding. SPE is formulated as a semidefinite program that learns a lowrank kernel matrix constrained by a set of linear inequalities which captures the connectivity structure of the input graph. Traditional graph embedding algorithms do not preserve structure according to our definition, and thus the resulting visualizations can be misleading or less informative. SPE provides significant improvements in terms of visualization and lossless compression of graphs, outperforming popular methods such as spectral embedding and Laplacian eigenmaps. We find that many classical graphs and networks can be properly embedded using only a few dimensions. Furthermore, introducing structure preserving constraints into dimensionality reduction algorithms produces more accurate representations of highdimensional data. 1.
Efficient Algorithms Using The Multiplicative Weights Update Method
, 2006
"... Abstract Algorithms based on convex optimization, especially linear and semidefinite programming, are ubiquitous in Computer Science. While there are polynomial time algorithms known to solve such problems, quite often the running time of these algorithms is very high. Designing simpler and more eff ..."
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Abstract Algorithms based on convex optimization, especially linear and semidefinite programming, are ubiquitous in Computer Science. While there are polynomial time algorithms known to solve such problems, quite often the running time of these algorithms is very high. Designing simpler and more efficient algorithms is important for practical impact. In this thesis, we explore applications of the Multiplicative Weights method in the design of efficient algorithms for various optimization problems. This method, which was repeatedly discovered in quite diverse fields, is an algorithmic technique which maintains a distribution on a certain set of interest, and updates it iteratively by multiplying the probability mass of elements by suitably chosen factors based on feedback obtained by running another algorithm on the distribution. We present a single metaalgorithm which unifies all known applications of this method in a common framework. Next, we generalize the method to the setting of symmetric matrices rather than real numbers. We derive the following applications of the resulting Matrix Multiplicative Weights algorithm: 1. The first truly general, combinatorial, primaldual method for designing efficient algorithms for semidefinite programming. Using these techniques, we obtain significantly faster algorithms for obtaining O(plog n) approximations to various graph partitioning problems, such as Sparsest Cut, Balanced Separator in both directed and undirected weighted graphs, and constraint satisfaction problems such as Min UnCut and Min 2CNF Deletion. 2. An ~O(n3) time derandomization of the AlonRoichman construction of expanders using Cayley graphs. The algorithm yields a set of O(log n) elements which generates an expanding Cayley graph in any group of n elements. 3. An ~O(n3) time deterministic O(log n) approximation algorithm for the quantum hypergraph covering problem. 4. An alternative proof of a result of Aaronson that the flfatshattering dimension of quantum states on n qubits is O ( nfl2).