 
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 DAGDirected Acyclic Graph G = (V, E) (directed graph with no directed cycles)
with nonnegative 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 nonnegative 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.
