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Math. Program. 01/2011; 126:315-350.
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ABSTRACT: We introduce a new approach to computing an approximately maximum s-t flow in a capacitated, undirected graph. This flow is computed by solving a sequence of electrical flow problems. Each electrical flow is given by the solution of a system of linear equations in a Laplacian matrix, and thus may be approximately computed in nearly-linear time. Using this approach, we develop the fastest known algorithm for computing approximately maximum s-t flows. For a graph having n vertices and m edges, our algorithm computes a (1-\epsilon)-approximately maximum s-t flow in time \tilde{O}(mn^{1/3} \epsilon^{-11/3}). A dual version of our approach computes a (1+\epsilon)-approximately minimum s-t cut in time \tilde{O}(m+n^{4/3}\eps^{-8/3}), which is the fastest known algorithm for this problem as well. Previously, the best dependence on m and n was achieved by the algorithm of Goldberg and Rao (J. ACM 1998), which can be used to compute approximately maximum s-t flows in time \tilde{O}(m\sqrt{n}\epsilon^{-1}), and approximately minimum s-t cuts in time \tilde{O}(m+n^{3/2}\epsilon^{-3}).
10/2010;
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ABSTRACT: We present a method for proving upper bounds on the eigenvalues of the graph
Laplacian. A main step involves choosing an appropriate "Riemannian" metric to
uniformize the geometry of the graph. In many interesting cases, the existence
of such a metric is shown by examining the combinatorics of special types of
flows. This involves proving new inequalities on the crossing number of graphs.
In particular, we use our method to show that for any positive integer k, the
kth smallest eigenvalue of the Laplacian on an n-vertex, 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 square planar grids. We also extend this
spectral result to graphs with bounded genus, and graphs which forbid fixed
minors. Previously, such spectral upper bounds were only known for the case
k=2.
08/2010;
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Commun. ACM. 01/2009; 52:76-84.
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50th Annual IEEE Symposium on Foundations of Computer Science, FOCS 2009, October 25-27, 2009, Atlanta, Georgia, USA; 01/2009
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ABSTRACT: We study the design of local algorithms for massive graphs. A local algorithm is one that finds a solution containing or near a given vertex without looking at the whole graph. We present a local clustering algorithm. Our algorithm finds a good cluster--a subset of vertices whose internal connections are significantly richer than its external connections--near a given vertex. The running time of our algorithm, when it finds a non-empty local cluster, is nearly linear in the size of the cluster it outputs. Our clustering algorithm could be a useful primitive for handling massive graphs, such as social networks and web-graphs. As an application of this clustering algorithm, we present a partitioning algorithm that finds an approximate sparsest cut with nearly optimal balance. Our algorithm takes time nearly linear in the number edges of the graph. Using the partitioning algorithm of this paper, we have designed a nearly-linear time algorithm for constructing spectral sparsifiers of graphs, which we in turn use in a nearly-linear time algorithm for solving linear systems in symmetric, diagonally-dominant matrices. The linear system solver also leads to a nearly linear-time algorithm for approximating the second-smallest eigenvalue and corresponding eigenvector of the Laplacian matrix of a graph. These other results are presented in two companion papers.
10/2008;
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ABSTRACT: We introduce a new notion of graph sparsificaiton based on spectral similarity of graph Laplacians: spectral sparsification requires that the Laplacian quadratic form of the sparsifier approximate that of the original. This is equivalent to saying that the Laplacian of the sparsifier is a good preconditioner for the Laplacian of the original. We prove that every graph has a spectral sparsifier of nearly linear size. Moreover, we present an algorithm that produces spectral sparsifiers in time $\softO{m}$, where $m$ is the number of edges in the original graph. This construction is a key component of a nearly-linear time algorithm for solving linear equations in diagonally-dominant matrcies. Our sparsification algorithm makes use of a nearly-linear time algorithm for graph partitioning that satisfies a strong guarantee: if the partition it outputs is very unbalanced, then the larger part is contained in a subgraph of high conductance. Comment: This revision addresses comments of the referees. In particular, we have completely re-written the proof of the main graph partitioning theorem in section 8
08/2008;
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Int. J. Comput. Geometry Appl. 01/2007; 17:1-30.
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ABSTRACT: We present a randomized algorithm that, on input a symmetric, weakly
diagonally dominant n-by-n matrix A with m nonzero entries and an n-vector b,
produces a y such that $\norm{y - \pinv{A} b}_{A} \leq \epsilon \norm{\pinv{A}
b}_{A}$ in expected time $O (m \log^{c}n \log (1/\epsilon)),$ for some constant
c. By applying this algorithm inside the inverse power method, we compute
approximate Fiedler vectors in a similar amount of time. The algorithm applies
subgraph preconditioners in a recursive fashion. These preconditioners improve
upon the subgraph preconditioners first introduced by Vaidya (1990).
For any symmetric, weakly diagonally-dominant matrix A with non-positive
off-diagonal entries and $k \geq 1$, we construct in time $O (m \log^{c} n)$ a
preconditioner B of A with at most $2 (n - 1) + O ((m/k) \log^{39} n)$ nonzero
off-diagonal entries such that the finite generalized condition number
$\kappa_{f} (A,B)$ is at most k, for some other constant c.
In the special case when the nonzero structure of the matrix is planar the
corresponding linear system solver runs in expected time $ O (n \log^{2} n + n
\log n \ \log \log n \ \log (1/\epsilon))$.
We hope that our introduction of algorithms of low asymptotic complexity will
lead to the development of algorithms that are also fast in practice.
07/2006;
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ABSTRACT: We prove that every weighted graph contains a spanning tree subgraph of average stretch O((log n log log n)^2). Moreover, we show how to construct such a tree in time O(m log^2 n).
12/2004;
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Euro-Par 2004 Parallel Processing, 10th International Euro-Par Conference, Pisa, Italy, August 31-September 3, 2004, Proceedings; 01/2004
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SPAA 2004: Proceedings of the Sixteenth Annual ACM Symposium on Parallelism in Algorithms and Architectures, June 27-30, 2004, Barcelona, Spain; 01/2004
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ABSTRACT: We present a linear-system solver that, given an $n$-by-$n$ symmetric positive semi-definite, diagonally dominant matrix $A$ with $m$ non-zero entries and an $n$-vector $\bb $, produces a vector $\xxt$ within relative distance $\epsilon$ of the solution to $A \xx = \bb$ in time $O (m^{1.31} \log (n \kappa_{f} (A)/\epsilon)^{O (1)})$, where $\kappa_{f} (A)$ is the log of the ratio of the largest to smallest non-zero eigenvalue of $A$. In particular, $\log (\kappa_{f} (A)) = O (b \log n)$, where $b$ is the logarithm of the ratio of the largest to smallest non-zero entry of $A$. If the graph of $A$ has genus $m^{2\theta}$ or does not have a $K_{m^{\theta}} $ minor, then the exponent of $m$ can be improved to the minimum of $1 + 5 \theta $ and $(9/8) (1+\theta)$. The key contribution of our work is an extension of Vaidya's techniques for constructing and analyzing combinatorial preconditioners.
11/2003;
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ABSTRACT: Let $\orig{A}$ be any matrix and let $A$ be a slight random perturbation of $\orig{A}$. We prove that it is unlikely that $A$ has large condition number. Using this result, we prove it is unlikely that $A$ has large growth factor under Gaussian elimination without pivoting. By combining these results, we bound the smoothed precision needed by Gaussian elimination without pivoting. Our results improve the average-case analysis of Gaussian elimination without pivoting performed by Yeung and Chan (SIAM J. Matrix Anal. Appl., 1997).
11/2003;
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ABSTRACT: This paper has been divided into three papers. arXiv:0809.3232, arXiv:0808.4134, arXiv:cs/0607105 Comment: withdrawn by author
10/2003;
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ABSTRACT: In smoothed analysis, one measures the complexity of algorithms assuming that their inputs are subject to small amounts of
random noise. In an earlier work (Spielman and Teng, 2001), we introduced this analysis to explain the good practical behavior
of the simplex algorithm. In this paper, we provide further motivation for the smoothed analysis of algorithms, and develop
models of noise suitable for analyzing the behavior of discrete algorithms. We then consider the smoothed complexities of
testing some simple graph properties in these models.
09/2003: pages 256-270;
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03/2003;
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ABSTRACT: We perform a smoothed analysis of the termination phase of an interior-point method. By combining this analysis with the smoothed analysis of Renegar's interior-point algorithm by Dunagan, Spielman and Teng, we show that the smoothed complexity of an interior-point algorithm for linear programming is $O (m^{3} \log (m/\sigma))$. In contrast, the best known bound on the worst-case complexity of linear programming is $O (m^{3} L)$, where $L$ could be as large as $m$. We include an introduction to smoothed analysis and a tutorial on proof techniques that have been useful in smoothed analyses.
02/2003;
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Math. Program. 01/2003; 97:375-404.
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Algorithms and Data Structures, 8th International Workshop, WADS 2003, Ottawa, Ontario, Canada, July 30 - August 1, 2003, Proceedings; 01/2003
