The solution of some random NP-hard problems in polynomial expected time
Journal of Algorithms
On the second eigenvalue of random regular graphs
STOC '89 Proceedings of the twenty-first annual ACM symposium on Theory of computing
Partitioning sparse matrices with eigenvectors of graphs
SIAM Journal on Matrix Analysis and Applications
A Spectral Technique for Coloring Random 3-Colorable Graphs
SIAM Journal on Computing
The metropolis algorithm for graph bisection
Discrete Applied Mathematics
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proceedings of the eighth international conference on Random structures and algorithms
STOC '01 Proceedings of the thirty-third annual ACM symposium on Theory of computing
Analysis of gene expression profiles: class discovery and leaf ordering
Proceedings of the sixth annual international conference on Computational biology
Heuristics for semirandom graph problems
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General partitioning on random graphs
Journal of Algorithms
A spectral technique for random satisfiable 3CNF formulas
SODA '03 Proceedings of the fourteenth annual ACM-SIAM symposium on Discrete algorithms
Coloring Bipartite Hypergraphs
Proceedings of the 5th International IPCO Conference on Integer Programming and Combinatorial Optimization
Spectral partitioning works: planar graphs and finite element meshes
FOCS '96 Proceedings of the 37th Annual Symposium on Foundations of Computer Science
Spectral Partitioning of Random Graphs
FOCS '01 Proceedings of the 42nd IEEE symposium on Foundations of Computer Science
Graph partitioning for high-performance scientific simulations
Sourcebook of parallel computing
The Largest Eigenvalue of Sparse Random Graphs
Combinatorics, Probability and Computing
On clusterings: Good, bad and spectral
Journal of the ACM (JACM)
Max Cut for Random Graphs with a Planted Partition
Combinatorics, Probability and Computing
Spectral Analysis of Random Graphs with Skewed Degree Distributions
FOCS '04 Proceedings of the 45th Annual IEEE Symposium on Foundations of Computer Science
A spectral heuristic for bisecting random graphs
SODA '05 Proceedings of the sixteenth annual ACM-SIAM symposium on Discrete algorithms
Spectral techniques applied to sparse random graphs
Random Structures & Algorithms
Spectral clustering by recursive partitioning
ESA'06 Proceedings of the 14th conference on Annual European Symposium - Volume 14
Eigenvalues and graph bisection: An average-case analysis
SFCS '87 Proceedings of the 28th Annual Symposium on Foundations of Computer Science
On the hardness and easiness of random 4-SAT formulas
ISAAC'04 Proceedings of the 15th international conference on Algorithms and Computation
Statistical algorithms and a lower bound for detecting planted cliques
Proceedings of the forty-fifth annual ACM symposium on Theory of computing
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In this paper we study the use of spectral techniques for graph partitioning. Let G = (V, E) be a graph whose vertex set has a ‘latent’ partition V1,. . ., Vk. Moreover, consider a ‘density matrix’ Ɛ = (Ɛvw)v, sw∈V such that, for v ∈ Vi and w ∈ Vj, the entry Ɛvw is the fraction of all possible Vi−Vj-edges that are actually present in G. We show that on input (G, k) the partition V1,. . ., Vk can (very nearly) be recovered in polynomial time via spectral methods, provided that the following holds: Ɛ approximates the adjacency matrix of G in the operator norm, for vertices v ∈ Vi, w ∈ Vj ≠ Vi the corresponding column vectors Ɛv, Ɛw are separated, and G is sufficiently ‘regular’ with respect to the matrix Ɛ. This result in particular applies to sparse graphs with bounded average degree as n = #V → ∞, and it has various consequences on partitioning random graphs.