An approximation algorithm for the generalized assignment problem

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 pif and incurs a cost of c,f, each machine / is available for 7", t ime units, and the objective is.t»minimize the total cost incurred. Our 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 / is used for at most 27", time units. We also extend this result to a variant of the problem where, instead of a fixed processing time p,r there is a range of possible processing times for each machine-job pair, and the cost linearly increases as (he processing time decreases. We 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. We also consider the objective of minimizing the mean job completion time. We show that there is a polynomial-time algorithm that, given values M and 7", either proves that no schedule of mean job completion time M and makespan /"exists, or else finds a schedule of mean job completion time at most M and makespan at most 27".