Approximation Schemes for Makespan Minimization

Years and Authors of Summarized Original Work

• 1987, 1988; Hochbaum, Shmoys

Problem Definition

Non-preemptive makespan minimization on m uniformly related machines is defined as follows. We are given a set M = { 1, 2, …, m} of m machines where each machine i has a speed s _i such that s _i  > 0. In addition we are given a set of jobs J = { 1, 2, …, n}, where each job j has a positive size p _j and all jobs are available for processing at time 0. The jobs need to be partitioned into m subsets S₁, …, S _m , with S _i being the subset of jobs assigned to machine i, and each such (ordered) partition is a feasible solution to the problem. Processing job j on machine i takes $\frac{p_{j}} {s_{i}}$ time units. For such a solution (also known as a schedule), we let $L_{i} = (\sum _{j\in S_{i}}p_{j})/s_{i}$ be the completion time or load of machine i. The work of machine i is $W_{i} =\sum _{j\in S_{i}}p_{j} = L_{i} \cdot s_{i}$, that is, the total size of the jobs assigned to i. The makespan...