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 - 2010 |
Externally published | Yes |
All Science Journal Classification (ASJC) codes
- General Computer Science
- General Mathematics
Keywords
- Grothendieck's inequality
- Laplacian eigenvalues
- Quasi-random graphs
- Regularity lemma