Analysis of some Krylov subspace approximations to the matrix exponential operator
SIAM Journal on Numerical Analysis
Cut problems and their application to divide-and-conquer
Approximation algorithms for NP-hard problems
On Krylov Subspace Approximations to the Matrix Exponential Operator
SIAM Journal on Numerical Analysis
The complexity of the matrix eigenproblem
STOC '99 Proceedings of the thirty-first annual ACM symposium on Theory of computing
Stochastic processes
On clusterings-good, bad and spectral
FOCS '00 Proceedings of the 41st Annual Symposium on Foundations of Computer Science
Graph partitioning for high-performance scientific simulations
Sourcebook of parallel computing
Expander flows, geometric embeddings and graph partitioning
STOC '04 Proceedings of the thirty-sixth annual ACM symposium on Theory of computing
STOC '04 Proceedings of the thirty-sixth annual ACM symposium on Theory of computing
Preconditioning Lanczos Approximations to the Matrix Exponential
SIAM Journal on Scientific Computing
Graph partitioning using single commodity flows
Proceedings of the thirty-eighth annual ACM symposium on Theory of computing
Local Graph Partitioning using PageRank Vectors
FOCS '06 Proceedings of the 47th Annual IEEE Symposium on Foundations of Computer Science
A combinatorial, primal-dual approach to semidefinite programs
Proceedings of the thirty-ninth annual ACM symposium on Theory of computing
An algorithm for improving graph partitions
Proceedings of the nineteenth annual ACM-SIAM symposium on Discrete algorithms
Optimal hierarchical decompositions for congestion minimization in networks
STOC '08 Proceedings of the fortieth annual ACM symposium on Theory of computing
On partitioning graphs via single commodity flows
STOC '08 Proceedings of the fortieth annual ACM symposium on Theory of computing
Finding sparse cuts locally using evolving sets
Proceedings of the forty-first annual ACM symposium on Theory of computing
Breaking the Multicommodity Flow Barrier for O(vlog n)-Approximations to Sparsest Cut
FOCS '09 Proceedings of the 2009 50th Annual IEEE Symposium on Foundations of Computer Science
Approaching Optimality for Solving SDD Linear Systems
FOCS '10 Proceedings of the 2010 IEEE 51st Annual Symposium on Foundations of Computer Science
Fast Approximation Algorithms for Cut-Based Problems in Undirected Graphs
FOCS '10 Proceedings of the 2010 IEEE 51st Annual Symposium on Foundations of Computer Science
Approximating Semidefinite Packing Programs
SIAM Journal on Optimization
IPCO'05 Proceedings of the 11th international conference on Integer Programming and Combinatorial Optimization
Proceedings of the twenty-second annual ACM-SIAM symposium on Discrete Algorithms
Algorithms, graph theory, and the solution of laplacian linear equations
ICALP'12 Proceedings of the 39th international colloquium conference on Automata, Languages, and Programming - Volume Part II
A simple, combinatorial algorithm for solving SDD systems in nearly-linear time
Proceedings of the forty-fifth annual ACM symposium on Theory of computing
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We give a novel spectral approximation algorithm for the balanced (edge-)separator problem that, given a graph G, a constant balance b ∈ (0,1/2], and a parameter γ, either finds an Ω(b)-balanced cut of conductance O(√γ) in G, or outputs a certificate that all b-balanced cuts in G have conductance at least γ, and runs in time ~O(m). This settles the question of designing asymptotically optimal spectral algorithms for balanced separator. Our algorithm relies on a variant of the heat kernel random walk and requires, as a subroutine, an algorithm to compute exp(-L)v where L is the Laplacian of a graph related to G and v is a vector. Algorithms for computing the matrix-exponential-vector product efficiently comprise our next set of results. Our main result here is a new algorithm which computes a good approximation to exp(-A)v for a class of symmetric positive semidefinite (PSD) matrices A and a given vector v, in time roughly ~O(mA), independent of the norm of A, where mA is the number of non-zero entries of A. This uses, in a non-trivial way, the result of Spielman and Teng on inverting symmetric and diagonally-dominant matrices in ~O(mA) time. Finally, using old and new uniform approximations to e-x we show how to obtain, via the Lanczos method, a simple algorithm to compute exp(-A)v for symmetric PSD matrices that runs in time roughly O(tA⋅ √norm(A)), where tA is the time required for the computation of the vector Aw for given vector w. As an application, we obtain a simple and practical algorithm, with output conductance O(√γ), for balanced separator that runs in time O(m/√γ). This latter algorithm matches the running time, but improves on the approximation guarantee of the Evolving-Sets-based algorithm by Andersen and Peres for balanced separator.