Unformatted text preview: C i is of the form MAJ ( x i 1 ,x i 2 ,x i 3 ) where z i is a boolean variable and MAJ is the majority function (the majority of its three literals’ values is 1). The problem asks to ﬁnd a truth assignment for all variables so as to maximize the number of satisﬁed clauses. Problem 3 . Prove the following: Let B ,B 1 , ··· ,B t be subsets of V such that β i =  B i  /n . Deﬁne ( B,t ) to be the event that a random walk ( v ,v 1 , ··· ,v t ) has the property that ∀ i, v i ∈ B i , we have: Pr [( B,t )] ≤ t1 Y i =0 ( p β i β i +1 + α ) Problem 4 . Prove the following: Let G = ( V,E ) be an ( n,d,α )graph (as deﬁned in class), then for every S,T ⊆ V , we have: ± ± ± ± d  S  T  nE ( S,T ) ± ± ± ± ≤ αd p  S  T  Problem 5 . Find the spectrum of a dregular tree of height h where d and h are given positive integers....
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 Spring '08
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 Natural number, NPcomplete, edge disjoint paths

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