Online Minimization of the Maximum Starting Time: Migration Helps

We consider non-preemptive load balancing on m identical machines where the cost of a machine is defined as the maximum starting time of a job assigned to the machine, and the goal is to find a partition of the jobs that minimizes the maximum machine cost. In our variant the last job on each machine is the smallest job assigned to that machine. The online model for this problem is too restrictive as a trivial example shows that there is no competitive algorithm for the problem. We show that a constant migration factor is sufficient to guarantee a (32+ε)-competitive algorithm for all ε> 0 , and using a constant migration factor cannot lead to a better than a 32-competitive algorithm. We also show that for this problem, constant amortized migration factor is strictly more powerful and allows us to obtain a polynomial time approximation scheme with a constant amortized migration factor. Thus, the ability to move some limited set of jobs on each step allows the algorithm to be much better than in the pure online settings.