Final - C i is of the form MAJ x i 1,x i 2,x i 3 where z i...

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CIS6930: PCPs and Inapproximability - Final Exam Due at 4pm on 12-08-2009 via email. As usual, I will NOT accept any late submission. This is the final exam. Therefore, no collaboration and discussion between students is allowed. No references except lecture notes is accepted. Do the following 5 required problems, 10 pts each: Problem 1 . Prove that it is NP-hard to approximate the Edge Disjoint Paths (EDP) problem within a factor of m 1 / 2 - ± for any ± > 0 where the EDP problem is defined as follows: Definition 1 Given a directed graph G = ( V,E ) with m = | E | and source-sink pairs ( s i ,t i ) for i = 1 ,...,t , find the maximum number of edge disjoint paths (paths that do not share edges) to connect these source-sink pairs (that is, our goal is to connect as many pairs as possible using edge disjoint paths) Problem 2. Prove that there is no (2 / 3+ ± )-approximation for MAX-3MAJ unless P = NP where MAX-3MAJ is defined as follows: Definition 2 An instance of MAX-3MAJ is a collection of m clauses in which each clause
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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 )] ≤ t-1 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 | n-E ( S,T ) ± ± ± ± ≤ αd p | S || T | Problem 5 . Find the spectrum of a d-regular tree of height h where d and h are given positive integers....
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This note was uploaded on 05/20/2011 for the course CIS 6930 taught by Professor Staff during the Spring '08 term at University of Florida.

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