Lecture 9 : The PCP Theorem Via gap Amplification
نویسندگان
چکیده
We start by presenting a PCP system with two queries as a graph. The vertices correspond to locations of the proof. The edges correspond to tests made by the verifier on the two endpoints. Each vertex is assigned a symbol from an alphabet Σ by the prover. Each edge is associated with a constraint on pairs in Σ×Σ. Note that the definition of “degree” we had before coincides with the definition of the maximal degree in the graph. We consider the following gap problem: given a graph as above, distinguish between the following two cases:
منابع مشابه
Cse 533: the Pcp Theorem and Hardness of Approximation Lecture 4: Pcp Theorem Proof: Degree Reduction, Expanderization
1 Constraint graphs and the PCP Theorem Definition 1.1. A constraint graph (G, C) over alphabet Σ is defined to be an undirected graph G = (V, E) together with a set of " constraints " C, one per edge. The constraint specified for edge e ∈ E, call it c e ∈ C, is just a subset of Σ × Σ. Note that the graph is allowed to have multi-edges (representing multiple constraints over the same pair of ve...
متن کاملPCP theorem: gap amplification
In this note, we prove the following theorem, which is a major step toward a proof of the PCP theorem. Recall that in the optimization problem 2-CSPW, we are given a list of arity-2 constraints between variables over an alphabet of size W and the goal is to find an assignment to the variables that satisfies as many of the constraints as possible. A 2-CSPW instance φ is d-regular if every variab...
متن کاملCS 286 . 2 Lecture 4 : Dinur ’ s Proof of the PCP Theorem
Previously, we have proven a weak version of the PCP theorem: NP ⊆ PCP1,1/2(r = poly, q = O(1)). With this result we have the desired constant completeness and soundness, but the proof is exponentially long. It takes difficult work to shorten the proof to get the full PCP theorem: NP ⊆ PCP1,1/2(r = O(log n), q = O(1)). Now we will see a different approach to the proof of the PCP theorem which w...
متن کاملLecture 2 : PCP Theorem and GAP - SAT ; intro to expanders
In the last lecture we said we will prove NP-hardness of approximation algorithms. To do this, we will follow the same approach that is used in NP-completeness — we convert an optimization problem into a decision problem. For a concrete example recall the MAX-3ESAT problem: each clause of the input boolean formula is an OR of exactly three literals and the goal is to find an assignment that max...
متن کاملLecture 17 : Hardness of Approximation
In this and the next few lectures we revisit the PCP Theorem with the aim of establishing tight inapproximability results for some natural NP-hard optimization problems. Earlier, we discussed the use of harmonic analysis in the derivation of the PCP Theorem. More specifically, in lecture 5 we devised tests for the Hadamard code and the long code, which play a critical role in the construction o...
متن کاملذخیره در منابع من
با ذخیره ی این منبع در منابع من، دسترسی به آن را برای استفاده های بعدی آسان تر کنید
عنوان ژورنال:
دوره شماره
صفحات -
تاریخ انتشار 2011