## The Importance of Being Biased (2002)

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Citations: | 84 - 7 self |

### BibTeX

@INPROCEEDINGS{Dinur02theimportance,

author = {Irit Dinur and Samuel Safra},

title = {The Importance of Being Biased},

booktitle = {},

year = {2002},

pages = {33--42}

}

### Years of Citing Articles

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### Abstract

The Minimum Vertex Cover problem is the problem of, given a graph, finding a smallest set of vertices that touches all edges. We show that it is NP-hard to approximate this problem 1.36067, improving on the previously known hardness result for a 6 factor. 1

### Citations

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Citation Context ...√ 5 − 21. Background. Let us now very briefly describe the background related to PCP, hardness of approximation, and reductions utilizing one to obtain the other. Vertex-Cover is in fact APX-compl=-=ete [PY91]-=-, namely, it belongs to a class of problems whose hardness of approximation is interrelated. All of the problems in this class (that includes the problem of finding the maximal cut in a given graph, f... |

366 | Probabilistic checking of proofs: A new characterization of NP
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Citation Context ...is a ’no’ instance and not even an arbitrarily small ɛ = 1 |R| O(1) fraction of the local-constraints of Φ can be satisfied. The next step calls for applying a version of the Composition techniq=-=ue of [AS92], as-=- proposed in [BGS98], to this specific setting. Namely, one replaces each of the variables with a set of variables representing its encoding, and each local-constraint ϕ ∈ Φ with constraints that ... |

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Citation Context ...te analysis. One of the most successful recipes for such reductions, is the scheme of [BGS98, H˚as99, H˚as97], whose rough sketch is as follows. The starting point is the parallel repetition lemma o=-=f [Raz98],-=- applied to the gap-SAT of [AS92, ALM + 92]. This demonstrates that, given a gap-SAT instance Φ, consisting of localconstraints, each over two variables whose range is R; it is NP-hard to decide betw... |

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Citation Context ...hat satisfies all Φ, and the case where any assignment A satisfies at most a small fraction of Φ. The FGLSS [FGL + 91, Kar72] reduction, applied to the PCP characterization of NP of either [Raz98] o=-=r [RS97], show-=-s the Independent-Set problem on a co-partite graph to be NP-hard. We introduce a slightly stronger version of that gap-Independent-Set problem – in which in the ’no’ case there is not even a si... |

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Citation Context ...µ p. For a proof of such a bound we venture into the field of extremal set theory, where maximal intersecting families have been studied for some time. This study was initiated by Erdős, Ko, and Rad=-=o [EKR61],-=- and has seen various extensions and generalizations. The corollary above is a generalization to µ p of what is known as the Complete Intersection Theorem for finite sets, that was proven by [AK97]. ... |

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Citation Context ...ing ∆ = Êi1 ∩ Êi2 , 1. Ĉi1 ∩ ∆ = Ĉi2 ∩ ∆ 2. ˆ F ♭ i1 ∩ ∆ = ˆ F ♭ ∩ ∆ i2 3. ˆ F ♯ i1 ∩ ∆ = ˆ F ♯ ∩ ∆ i2 Proof: Our proof begins by applying the following Sun=-=flower-Lemma over the sets Êi: Theorem 6.8 ([ER60]) There exists some -=-integer function Γ∗(k, d) (not depending on |R|), such that for any F ⊂ � � R k , if |F| ≥ Γ∗(k, d), there are d distinct sets F1, . . . , Fd ∈ F, such that, let ∆ def = F1 ∩ . . .... |

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Citation Context ...not even an arbitrarily small ɛ = 1 |R| O(1) fraction of the local-constraints of Φ can be satisfied. The next step calls for applying a version of the Composition technique of [AS92], as proposed i=-=n [BGS98], to-=- this specific setting. Namely, one replaces each of the variables with a set of variables representing its encoding, and each local-constraint ϕ ∈ Φ with constraints that both verify the consiste... |

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The complete intersection theorem for systems of finite sets
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Citation Context ... [EKR61], and has seen various extensions and generalizations. The corollary above is a generalization to µ p of what is known as the Complete Intersection Theorem for finite sets, that was proven by=-= [AK97]. Frankl [Fr-=-a78] defined the following families: Ai,t def = {F ∈ P ([n]) | F ∩ [1, t + 2i] ≥ t + i} which are easily seen to be t-intersecting for 0 ≤ i ≤ n−t 2 ; and conjectured the following theorem... |

69 | Clique is hard to approximate within n 1 - Hastad - 1999 |

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- Friedgut
- 1998
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Citation Context ...ce a bounded continuous function cannot have a large derivative everywhere, Russo’s Lemma guarantees that f will have low average-sensitivity. At this point we can apply the insightful Friedgut Lemm=-=a [Fri98]-=- (Theorem 5.2) which essentially asserts that if a Boolean function has low average-sensitivity then it is almost entirely determined by only a small number of its variables. These variables make up t... |

59 | inapproximability results for maxclique, chromatic number and min-3lin-deletion - KHOT, PONNUSWAMI |

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37 | The exact bound in the Erdős-Ko-Rado theorem - Wilson - 1984 |

28 | Influences of variables and threshold intervals under group symmetries - Bourgain, Kalai - 1997 |

28 |
The Erdős-Ko-Rado theorem is true for n = ckt
- Frankl
- 1976
(Show Context)
Citation Context ...as seen various extensions and generalizations. The corollary above is a generalization to µ p of what is known as the Complete Intersection Theorem for finite sets, that was proven by [AK97]. Frankl=-= [Fra78] defined the-=- following families: Ai,t def = {F ∈ P ([n]) | F ∩ [1, t + 2i] ≥ t + i} which are easily seen to be t-intersecting for 0 ≤ i ≤ n−t 2 ; and conjectured the following theorem that was finall... |

26 | Approximation Algorithms for Steiner and Directed Multicuts
- Klein, Plotkin, et al.
- 1997
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Citation Context ...inding the minimum weight set of clauses in a 2-CNF formula whose deletion makes the formula satisfiable. The best approximation algorithm for this problem guarantees only a factor of log n log log n =-=[KPRT97]-=-. The framework for proving hardness results suggested herein can be tried on other problems for which the known hardness result does not match the best upper-bound. In essence one has to consider var... |

23 | An approximate zero-one law - Russo - 1982 |

13 | Collective coin flipping and other models of imperfect randomness - Ben-Or, Linial, et al. - 1987 |

11 | Clique is hard to approximate within n 1−o(1 - Engebretsen, Holmerin - 2000 |

9 | Some further approximation algorithms for the vertex cover problem - Monien, Speckenmeyer - 1983 |