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hw7520_11_2 - CS 7250 Approximation Algorithms Homework 2...

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CS 7250, Approximation Algorithms Homework 2 Tue, Feb 1, 2011 Due Tue, Feb 8, 2011 Problem 1: Greedy Vertex Cover Perhaps the first strategy one tries when designing an algorithm for an optimization problem is the greedy strategy. For the unweighted vertex cover problem, this would involve iteratively picking a maximum degree vertex and removing it, together with edges incident at it, until there are no edges left. (a) Show that this algorithm achieves an approximation guarantee of O (log n ). (b) Give a tight example: Class of input instances where this algorithm performs as bad as Ω(log n ). Problem 2: Set Coverage Maximum set coverage is the following problem: Given a set U of n elements, a collection of subsets of U , S 1 , . . . , S m , and an integer k , pick k sets so as to maximize the number of covered elements. (a) Show that maximum set coverage is NP-hard. (b) Show that the obvious algorithm, of greedily picking the best set in each iteration until k sets are picked, achieves an approximation factor of 1 - ( 1 - 1 k ) k > 1 - 1 e .
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