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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 π , such that V accepts with probability c ( n ). • If x 6∈ L then for all proofs π , V accepts with probability at most s ( n ). 2.2 Adaptive and non–adaptive verifiers One distinguishes between adaptive and non–adaptive verifiers. For adaptive verifier its i th query can depend on “the past” i.e. on the previous queries. For non–adaptive verifiers, the positions in the proof to be queried should be read simultaneously. By definition adaptive verifiers are stronger than non–adaptive 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 non–adaptive verifier using 2 q queries. Though less powerful, the non–adaptive verifiers have the advantage of being simpler to reason about and in fact the best known PCP constructions have non–adaptive 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 non–adaptive 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 { π e : { 1 ,...,k } t → { , 1 }} e ∈ E , the gap–generalized 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. 201 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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 Spring '09
 MadhuSudan
 Graph Theory, Bipartite graph, Graph coloring, Raz, H˚ astad

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