Approximation schemes for scheduling

We consider the classic scheduling/load balancing problems where there are m identical machines and n jobs, and each job should be assigned to some machine. Traditionally, the assignment of jobs to machines is measured by the makespan (maximum load) i.e., the L_∞ norm of the assignment. An ε-approximation scheme was given by Hochbaum and Shmoys for minimizing the L_∞ norm. In several applications, such as in storage allocation, a more appropriate measure is the sum of the squares of the loads (which is equivalent to the L₂ norm). This problem was considered in [4, 5, 13] who showed how to approximate the optimum value by a factor of about 1.04. In fact, a more general measure, which is the L_p norm (for any p≥1) can also be approximated to some constant which may be as large as 3/2. We improve these results by providing an ε-approximation scheme for the general L_p norm (and in particular for the L₂ norm). We also consider the case of restricted assignment of unit jobs where we show how to find in polynomial time, a solution which is optimal for all norms.