Scott, Alexander D. and Sorkin, Gregory B.
(2006)
*Solving sparse random instances of Max Cut and Max 2-CSP in linear expected time.*
Combinatorics, Probability and Computing, 15 (1-2).
pp. 281-315.
ISSN 0963-5483

## Abstract

We show that a maximum cut of a random graph below the giant-component threshold can be found in linear space and linear expected time by a simple algorithm. In fact, the algorithm solves a more general class of problems, namely binary 2-variable constraint satisfaction problems. In addition to Max Cut, such Max 2-CSPs encompass Max Dicut, Max 2-Lin, Max 2-Sat, Max-Ones-2-Sat, maximum independent set, and minimum vertex cover. We show that if a Max 2-CSP instance has an ‘underlying’ graph which is a random graph $G \in \mathcal{G}(n,c/n)$, then the instance is solved in linear expected time if $c \leq 1$. Moreover, for arbitrary values (or functions) $c>1$ an instance is solved in expected time $n \exp(O(1+(c-1)^3 n))$; in the ‘scaling window’ $c=1+\lambda n^{-1/3}$ with $\lambda$ fixed, this expected time remains linear. Our method is to show, first, that if a Max 2-CSP has a connected underlying graph with $n$ vertices and $m$ edges, then $O(n 2^{(m-n)/2})$ is a deterministic upper bound on the solution time. Then, analysing the tails of the distribution of this quantity for a component of a random graph yields our result. Towards this end we derive some useful properties of binomial distributions and simple random walks.

Item Type: | Article |
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Official URL: | http://journals.cambridge.org/action/displayJourna... |

Additional Information: | © 2006 Cambridge University Press |

Divisions: | Management |

Subjects: | Q Science > QA Mathematics |

Date Deposited: | 13 Apr 2011 13:51 |

Last Modified: | 20 Feb 2021 03:59 |

URI: | http://eprints.lse.ac.uk/id/eprint/35466 |

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