Trevisan, Luca, Sorkin, Gregory B., Sudan, Madhu and Williamson, David P.
(2000)
*Gadgets, approximation, and linear programming.*
SIAM Journal on Computing, 29 (6).
pp. 2074-2097.
ISSN 0097-5397

## Abstract

We present a linear programming-based method for finding "gadgets," i.e., combinatorial structures reducing constraints of one optimization problem to constraints of another. A key step in this method is a simple observation which limits the search space to a finite one. Using this new method we present a number of new, computer-constructed gadgets for several different reductions. This method also answers a question posed by Bellare, Goldreich, and Sudan [SIAM J. Comput., 27 (1998), pp. 804-915] of how to prove the optimality of gadgets: linear programming duality gives such proofs. The new gadgets, when combined with recent results of Håstad [Proceedings of the 29th ACM Symposium on Theory of Computing, 1997, pp. 1-10], improve the known inapproximability results for MAX CUT and MAX DICUT, showing that approximating these problems to within factors of 16/17+ ∊ and 12/13+ ∊ , respectively, is NP-hard for every ∊ > 0. Prior to this work, the best-known inapproximability thresholds for both problems were 71/72 (M. Bellare, O. Goldreich, and M. Sudan [SIAM J. Comput., 27 (1998), pp. 804-915]). Without using the gadgets from this paper, the best possible hardness that would follow from Bellare, Goldreich, and Sudan and Håstad is 18/19. We also use the gadgets to obtain an improved approximation algorithm for MAX3 SAT which guarantees an approximation ratio of .801.

Item Type: | Article |
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Official URL: | http://www.siam.org/journals/sicomp.php |

Additional Information: | © 2000 Society for Industrial and Applied Mathematics |

Divisions: | Management |

Subjects: | Q Science > QA Mathematics > QA75 Electronic computers. Computer science Q Science > QA Mathematics > QA76 Computer software |

Sets: | Research centres and groups > Management Science Group Departments > Management |

Date Deposited: | 13 Apr 2011 14:26 |

Last Modified: | 20 Jul 2021 00:45 |

URI: | http://eprints.lse.ac.uk/id/eprint/35498 |

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