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# hw2 - c&amp;gt 0 there exists a constant d&amp;gt 0...

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CIS6930: PCPs and Inapproximability - Homework 2 Due at the beginning of the lecture on 11-24-2009 . No late assignment will be accepted. Do the following 4 required problems, 10 pts each: Problem 1 . Using the gap-preserving reduction form Label Cover, show that for every ǫ > 0, there is no (1 - ǫ ) ln n approximation algorithm exists for the set cover problem unless NP DTIME ( n O (loglog n ) where n is the size of the universe. Problem 2 . Use the special graph to show Problem 2 (Saving Random Bits). [Look at the lecture note Expanders.pdf page 3 and 4.] Problem 3 . Prove Theorem 1 in the lecture note. (That is: For all
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Unformatted text preview: c &amp;gt; 0, there exists a constant d &amp;gt; 0 and n &amp;gt; 0 such that an ( n,d,c )-edge expander graph exits for all d d and n n .) Problem 4 . Let G be a d-regular graph and let 1 2 n be the eigenvalues as mentioned in the lecture note. Show the following: 1 = d and the corresponding eigenvector is x 1 = ( 1 / n ) T = (1 / n,. .., 1 / n ) T The graph is connected i 1 &amp;gt; 2 The graph is bipartite i 1 =- n 1...
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