 
Summary: Convex Programming for Scheduling Unrelated Parallel Machines
Yossi Azar # Amir Epstein +
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 p ij . The goal
is to find a schedule that minimizes the # p norm. Previous work showed a 2approximation algo
rithm for the problem with respect to the # # norm. For any fixed # p norm the previously known
approximation algorithm has a performance of #(p). We provide a 2approximation algorithm
for any fixed # p norm (p > 1). This algorithm uses convex programming relaxation. We also
give a # 2approximation algorithm for the # 2 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 # p norm a PTAS for any fixed number of machines. We also consider the
multidimensional generalization of the problem in which the jobs are ddimensional. Here the
goal is to minimize the # p 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 # p norm (p > 1).
1 Introduction
We consider the classical problem of scheduling jobs on parallel unrelated machines. Lenstra et. al
[14] and Shmoys and Tardos [16] provided a 2approximation algorithm for minimizing the makespan
