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Unformatted text preview: PCPs and Inapproxiability CIS 6930 Oct 29 th , 2009 Multicut and SparesestCut problem Lecturer: Dr. My T. Thai Scribe: Dung T. Nguyen 1 Unique Games Conjecture 1.1 The unique game Recall the twoprover one round proof system which can be seen as a game between two provers and a verifier. There are two sets of all possible ”questions” V and W that the verifier can ask the first and second prover respectively. The strategy of the first prover is a map L V : V → N where N is a set of all possible answers of the first prover. Given a question u , the first prover returns the answer L V ( u ) to the verifier. Similarly, the stratefy of the second prover is a map L W : W → M where M is the set of its possible answers. The decision of verifier is a map: Γ : V × N × W × M → { TRUE,FALSE } The game between the verifier and provers is described as follow. The verifier picks a pair of question ( v,w ), v ∈ V , w ∈ W with a certain probability dis tribution on the set of all pairs. It asks question v to the first prover, question w to the second one. Two provers return answers L V ( v ) ,L W ( w ) respectively. The verifier accepts iff : Γ( v,L V ( v ) ,w,L W ( w )) = TRUE The value of game is defined as the maximum, over all possible prover strategies, of acceptance probability of the verifier. Consider the game such that the answer of the second prover uniquely deter mines the answer of the first prover. It means that for every question pair ( v,w ) asked by the verifier and every answer b ∈ M of the second prover, there is a unique answer a ∈ N that makes the verifier accept. Thus, we can associate a function Π vw : M → N to the question pair ( v,w ) such that the verifier accepts iff: Π vw ( L W ( w )) = L V ( v ) Multicut and SparesestCut problem1 Figure 1 : Map functions from the answer of the second prover to the answer of the first prover The game is called unique if M = N and every function Π vw is a bijection. Following is an example of unique game. [1]: • The verifier sends a random bit to each prover • Each prover responds with a bit (so k=2 here) • The verifier accepts iff the XOR of the answers is equal to the AND of the questions The unique game can be represented on biparttitle graph G ( V,W,E ) where V = { q 1 1 ,q 1 2 ,...,q 1 n } and W = { q 2 1 ,q 2 1 ,...,q 2 n } are sets of questions for the first and second prover respectively. Set of answer is { 1 , 2 ,...,d } where d is the size of answer sets. Each edge ( q 1 i ,q 2 j ) ∈ E has weight w ij with the total weight is 1 and a associated bijection Π ij : [ d ] → [ d ] which maps every answer for the question q 1 i to a distinct answer for question q 2 j . Given an assignment A = { A p i  p ∈ [2] ,A p i ∈ [ d ] } of answers to questions, the edge ( q 1 i ,q 2 j ) is sastified if A 2 j = Π ij ( A 1 i ). The goal is to find an assignment that maximizes the total weight of sastified edges....
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 Spring '08
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 Graph Theory, Computational complexity theory, Quantification, Boolean function, Multicut, unique 2prover game

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