### Abstract

We deal with two intimately related subjects: quasi-randomness and regular partitions. The purpose of the concept of quasi-randomness is to express how much a given graph "resembles" a random one. Moreover, a regular partition approximates a given graph by a bounded number of quasi-random graphs. Regarding quasi-randomness, we present a new spectral characterization of low discrepancy, which extends to sparse graphs. Concerning regular partitions, we introduce a concept of regularity that takes into account vertex weights, and show that if G = (V, E) satisfies a certain boundedness condition, then G admits a regular partition. In addition, building on the work of Alon and Naor [Proceedings of the 36th ACM Symposium on Theory of Computing (STOC), Chicago, IL, ACM, New York, 2004, pp. 72-80], we provide an algorithm that computes a regular partition of a given (possibly sparse) graph G in polynomial time. As an application, we present a polynomial time approximation scheme for MAX CUT on (sparse) graphs without "dense spots."

Original language | English (US) |
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Pages (from-to) | 2336-2362 |

Number of pages | 27 |

Journal | SIAM Journal on Computing |

Volume | 39 |

Issue number | 6 |

DOIs | |

State | Published - May 19 2010 |

Externally published | Yes |

### All Science Journal Classification (ASJC) codes

- Computer Science(all)
- Mathematics(all)

### Keywords

- Grothendieck's inequality
- Laplacian eigenvalues
- Quasi-random graphs
- Regularity lemma

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## Cite this

*SIAM Journal on Computing*,

*39*(6), 2336-2362. https://doi.org/10.1137/070709529