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lect20 - 6.841 Advanced Complexity Theory April 22, 2009...

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Unformatted text preview: 6.841 Advanced Complexity Theory April 22, 2009 Lecture 20 Lecturer: Madhu Sudan Scribe: Kristian Brander 1 Introduction The purpose of todays lecture is to explore the landscape of PCPs. In particular three constructions of Dinur, Raz and H astad will be surveyed. 2 Recap from previous lectures 2.1 The PCP class A language L is in PCP c,s ( r,q ) if there exists a verifier V such that If x L there exists a proof , such that V accepts with probability c ( n ). If x 6 L then for all proofs , V accepts with probability at most s ( n ). 2.2 Adaptive and nonadaptive verifiers One distinguishes between adaptive and nonadaptive verifiers. For adaptive verifier its i th query can depend on the past i.e. on the previous queries. For nonadaptive verifiers, the positions in the proof to be queried should be read simultaneously. By definition adaptive verifiers are stronger than nonadaptive ones. An adaptive verifiers decision of which position to query next (given the past), may be represented as a decision tree, and therefore by querying all nodes at once, an adaptive verifier with q queries can be converted into a nonadaptive verifier using 2 q queries. Though less powerful, the nonadaptive verifiers have the advantage of being simpler to reason about and in fact the best known PCP constructions have nonadaptive verifiers. However the two types of verifiers really have different properties as the combination of the following results show. H astad in [H as01] and Guruswami, Lewin, Sudan and Trevisan in [GLST98] shows that NP PCP 1 , . 51 [ O (log n ) , 3] , which can be compared to a result of Trevisan and Zwick NAPCP 1 , . 51 ( O (log n, 3) P. Here NAPCP c,s ( r,q ) denotes the class of languages that has nonadaptive PCP verifiers. 2.3 Generalized graph coloring Let G with vertex set V and edges E V t where t is an integer. Given k colors and constraints { e : { 1 ,...,k } t { , 1 }} e E , the gapgeneralized hypergraph coloring problem GGHC c,s ( t,k ) is to find an assignment of colors A : V { 1 ,...,k } satisfying the constraints. UNSAT( G ) is the minimal fraction of unsatisfied edges for any assignment of colors. 20-1 For a given graph G , one may try to separate the two cases UNSAT( G ) 1- c and UNSAT( G ) 1- s , and this establishes a connection between hypergraph coloring and PCPs: To a proof one associates a hypergraph whose vertices are the bits of the proof and the hyperedges are queries. A proof can then be thought of as a coloring of the hypergraph, and the checks done by the verifier, as checking whether certain edge constraints...
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lect20 - 6.841 Advanced Complexity Theory April 22, 2009...

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