### Abstract

We consider a variation of the spectral sparsification problem where we are required to keep a subgraph of the original graph. Formally, given a union of two weighted graphs G and W and an integer k, we are asked to find a k-edge weighted graph W
_{k}
such that G+W
_{k}
is a good spectral sparsifer of G+W. We will refer to this problem as the subgraph (spectral) sparsification. We present a nontrivial condition on G and W such that a good sparsifier exists and give a polynomial-time algorithm to find the sparsifer. As an application of our technique, we show that for each positive integer k, every n-vertex weighted graph has an (n-1+k)-edge spectral sparsifier with relative condition number at most n/k log n, Õ(log log n) where Õ() hides lower order terms. Our bound nearly settles a question left open by Spielman and Teng about ultrasparsifiers. We also present another application of our technique to spectral optimization in which the goal is to maximize the algebraic connectivity of a graph (e.g. turn it into an expander) with a limited number of edges.

Original language | English (US) |
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Title of host publication | STOC'10 - Proceedings of the 2010 ACM International Symposium on Theory of Computing |

Pages | 57-65 |

Number of pages | 9 |

DOIs | |

State | Published - Jul 23 2010 |

Event | 42nd ACM Symposium on Theory of Computing, STOC 2010 - Cambridge, MA, United States Duration: Jun 5 2010 → Jun 8 2010 |

### Publication series

Name | Proceedings of the Annual ACM Symposium on Theory of Computing |
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ISSN (Print) | 0737-8017 |

### Other

Other | 42nd ACM Symposium on Theory of Computing, STOC 2010 |
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Country | United States |

City | Cambridge, MA |

Period | 6/5/10 → 6/8/10 |

### Keywords

- algebraic connectivity
- approximation algorithms
- graph sparsification
- ultrasparsifiers

### ASJC Scopus subject areas

- Software

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## Cite this

*STOC'10 - Proceedings of the 2010 ACM International Symposium on Theory of Computing*(pp. 57-65). (Proceedings of the Annual ACM Symposium on Theory of Computing). https://doi.org/10.1145/1806689.1806699