Algorithmic Methods Fall Semester, 2010/11 Final Exam: January 31, 2011 Summary: Algorithmic Methods Fall Semester, 2010/11 Final Exam: January 31, 2011 Lecturer: Prof. Yossi Azar Solve 4 out of the 5 questions. Write short but full and accurate answers. Each question should start on a new page and each of its parts should not exceed a page. No extra material is allowed. 1. We are given n jobs and m unrelated machines. The load of job i on machine j is wij. The load of a machine is the sum of the weights of the jobs assigned to it. In contrast to the standard problem here each job i has two copies and they should be assigned exactly to TWO different machines say j1 = j2 (then the load of j1 would increase by wij1 and the load j2 would increase by wij2 ). The goal is to minimize the maximum load. (a) Write the appropriate LP formulation. (b) Round the LP and provide a 2 approximation algorithm. (recall that the two machines each job is assigned to must be different) 2. We are given a DAG-Directed Acyclic Graph G = (V, E) (directed graph with no directed cycles) with non-negative weight we on each edge e E. The cost of increasing or decreasing the weight of an edge e by each unit is ce for each e E. One need to modify the weights (increase or decrease) such that for all vertices u, v V the lengths of any two paths from u to v (if exist) differ by a factor of at most 2. The weight of each edge must stay non-negative after the modification. The goal is to minimize total cost of the modification. (a) Form an LP for the problem and show how to solve it by a polynomial time algorithm. Collections: Computer Technologies and Information Sciences