CS 7250, Approximation AlgorithmsHomework 2Tue, Feb 1, 2011Due Tue, Feb 8, 2011Problem 1: Greedy Vertex CoverPerhaps the first strategy one tries when designing an algorithm for an optimization problem is thegreedy strategy. For the unweighted vertex cover problem, this would involve iteratively pickinga maximum degree vertex and removing it, together with edges incident at it, until there are noedges left.(a) Show that this algorithm achieves an approximation guarantee ofO(logn).(b) Give a tight example: Class of input instances where this algorithm performs as bad as Ω(logn).Problem 2: Set CoverageMaximum set coverage is the following problem: Given a setUofnelements, a collection of subsetsofU,S1, . . . , Sm, and an integerk, pickksets 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 untilksetsare picked, achieves an approximation factor of 1-(1-1k)k>1-1e.
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