Convex programming for scheduling unrelated parallel machines

Abstract

We consider the classical problem of scheduling parallel unrelated machines. Each job is to be processed by exactly one machine. Processing job j on machine i requires time pij. The goal is to find a schedule that minimizes the lp norm. Previous work showed a 2-approximation algorithm for the problem with respect to the l∞ norm. For any fixed lp norm the previously known approximation algorithm has a performance of θ(p). We provide a 2-approximation algorithm for any fixed lp norm (p>1). This algorithm uses convex programming relaxation. We also give a √ 2-approximation algorithm for the l2 norm. This algorithm relies on convex quadratic programming relaxation. To the best of our knowledge, this is the first time that general convex programming techniques (apart from SDPs and CQPs) are used in the area of scheduling. We show for any given lp norm a PTAS for any fixed number of machines. We also consider the multidimensional generalization of the problem in which the jobs are d-dimensional. Here the goal is to minimize the lp norm of the generalized load vector, which is a matrix where the rows represent the machines and the columns represent the jobs dimension. For this problem we give a (d+1)-approximation algorithm for any fixed lp norm (p>1).