## Expander Flows, Geometric Embeddings and Graph Partitioning (2004)

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Venue: | IN 36TH ANNUAL SYMPOSIUM ON THE THEORY OF COMPUTING |

Citations: | 237 - 18 self |

### BibTeX

@INPROCEEDINGS{Arora04expanderflows,,

author = {Sanjeev Arora and Satish Rao and Umesh Vazirani},

title = {Expander Flows, Geometric Embeddings and Graph Partitioning},

booktitle = {IN 36TH ANNUAL SYMPOSIUM ON THE THEORY OF COMPUTING},

year = {2004},

pages = {222--231},

publisher = {}

}

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### Abstract

We give a O( log n)-approximation algorithm for sparsest cut, balanced separator, and graph conductance problems. This improves the O(log n)-approximation of Leighton and Rao (1988). We use a well-known semidefinite relaxation with triangle inequality constraints. Central to our analysis is a geometric theorem about projections of point sets in , whose proof makes essential use of a phenomenon called measure concentration.

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Citation Context ...ent work. Semidefinite programming and approximation algorithms: Semidefinite programs (SDPs) have numerous applications in optimization. They are solvable in polynomial time via the ellipsoid method =-=[18]-=-, and 2 Note that this notion of graph embedding has no connection in general to the area of geometric embeddings. It is somewhat confusing that this paper features both notions and shows some connect... |

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Citation Context ...ation for graph conductance follows from the connection —first discovered in the context of Riemannian manifolds [8]—between conductance and the eigenvalue gap of the Laplacian: 2Φ(G) ≥ λ ≥ Φ(G) 2 /2 =-=[3, 2, 21]-=-. The approximation factor is 1/Φ(G), and hence Ω(n) intheworstcase, and O(1) only if Φ(G) is a constant. This connection between eigenvalues and expansion has had enormous influence in a variety of f... |

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Citation Context ... ∈ Ti such that 〈vj − vi,u〉≥ɛ1. Also for δ fraction of directions u, there is a point in vk ∈ Xi such that 〈vk − vi,u〉≤−ɛ and vk is matched to vi 4 Levy’s isoperimetric inequality is not trivial; see =-=[29]-=- for a sketch. However, results qualitatively the same —but with worse constants— as Lemma 9 can be derived from the more elementary Brunn-Minkowski inequality; this “approximate isoperimetric inequal... |

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