Instructor : Alina Ene Scribe : Alina Ene 1 Scheduling on Unrelated Parallel Machines

We have a set J of n jobs, and a set M of m machines. The processing time of job i is pij on machine j. Let f : J ←M be a function that assigns each job to exactly one machine. The makespan of f is max1≤j≤m ∑ i:f(i)=j pij , where ∑ i:f(i)=j pij is the total processing time of the jobs that are assigned to machine j. In the Scheduling on Unrelated Parallel Machines problem, the goal is to find an assignment of jobs to machines of minimum makespan. We can write an LP for the problem that is very similar to the routing LP from the previous lecture. For each job i and each machine j, we have a variable xij that denotes whether job i is assigned to machine j. We also have a variable λ for the makespan. We have a constraint for each job that ensures that the job is assigned to some machine, and we have a constraint for each machine that ensures that the total processing time of jobs assigned to the machines is at most the makespan λ.