Approximation schemes for the min-max starting time problem

We consider the off-line scheduling problem of minimizing the maximal starting time. The input to this problem is a sequence of n jobs and m identical machines. The goal is to assign the jobs to the machines so that the first time at which all jobs have already started running is minimized, under the restriction that the processing of the jobs on any given machine must respect their original order. Our main result is a polynomial time approximation scheme (PTAS) for this problem in the case where m is considered as part of the input. As the input to this problem is a sequence of jobs, rather than a set of jobs where the order is insignificant, we present techniques that are designed to handle order constraints imposed by the sequence. Those techniques are combined with common techniques of assignment problems in order to yield a PTAS for this problem. We also show that when m is a constant, the problem admits a fully polynomial time approximation scheme. Finally, we show that the makespan problem in the linear hierarchical model may be reduced to the min-max starting time problem, thus concluding that the former problem also admits a PTAS.