The generalized assignment problem can be viewed as the following problem of scheduling parallel machines with costs. Each job is to be processed by exactly one machine; processing job j on machine i requires time pij and incurs a cost of cij; each machine i is available for Ti time units, and the objective is to minimize the total cost incurred. The present main result is as follows. There is a polynomial-time algorithm that, given a value C, either proves that no feasible schedule of cost C exists, or else finds a schedule of cost at most C where each machine i is used for at most 2Ti time units. The authors also extend this result to a variant of the problem where, instead of a fixed processing time pij, there is a range of possible processing times for each machine-job pair, and the cost linearly increases as the processing time decreases. They show that these results imply a polynomial-time 2-approximation algorithm to minimize a weighted sum of the cost and the makespan, i.e., the maximum job completion time. The authors also consider the objective of minimizing the mean job completion time. They show that there is a polynomial-time algorithm that, given values M and T, either proves that no schedule of mean job completion time M and makespan T exists, or else finds a schedule of mean job completion time at most M and makespan at most 2T.
