hw7520_11_4

hw7520_11_4 - CS 7250, Approximation Algorithms Homework 4...

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CS 7250, Approximation Algorithms Homework 4 Thu, April 5, 2011 Due Tue, April 12, 2011 Problem 1 The performance guarantee α OPT, α > 0 . 878, for approximating MAXCUT was assuming that the vector program (26.2) on page 256 of Vazirani’s book can be solved optimally in polynomial time. In reality, such vector programs cannot be solved optimally in polynomial time, but they can be approximated to any desired degree of accuracy in polynomial time. Therefore, if ~ a i , 1 i n , is an oprimal solution for (26.2), suppose that we have instead an approximation ~ b i , 1 i n , with the guarantee that, for all i , (a)(1 - ± ) | ~ a i | ≤ | ~ b i | ≤ (1 + ± ) | ~ a i | and (b) ~ b i lies inside a cone of angle φ ± around the direction of ~ a i . How would that affect the statement of Lemma 26.9 ? You may assume that 0 ± << 1 / 100 and 0 φ ± << π/ 100. Problem 2 (a) In Vazirani’s book, for MAXCUT, Theorem 26.11 shows how to obtain a ”high probability statement” from Lemma 26.9. Obtain a similar statement for MAS-2SAT, starting from Lemma
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