Compact enumeration for scheduling one machine

Abstract

A basic strongly NP-hard scheduling problem in which non-simultaneously released jobs with delivery times are to be scheduled on a single machine with the objective to minimize the maximum job full completion time is considered. It is shown that only a relatively small group of jobs contribute in the complexity status of the problem, in the sense that if there were no such jobs the problem would be possible to solve in a polynomial time. These "critical" jobs are identified and the solution process is separated in two main phases so that at the first phase the remaining non-critical jobs are scheduled in a low degree polynomial time, and at the second phase an enumeration procedure for at most ν! permutations of the ν critical jobs is combined with a procedure that inserts these jobs according to the order in each of the permutations into the partial schedule of the non-critical jobs (this framework can, in general, be applied to a class of scheduling problems with job release and delivery times). Based on this framework, we describe an exact implicit enumeration algorithm (IEA) and a polynomial-time approximation scheme (PTAS) for the single-machine environment. Although the worst-case complexity analysis of IEA yields a factor of ν!, large sets of the permutations of the critical jobs can be discarded by incorporating a heuristic search strategy, in which the permutations of the critical jobs are considered in a special priority order. Not less importantly, in practice, the number ν turns out to be several times smaller than the total number of jobs n, and it becomes smaller when n increases. The above characteristics also apply to the proposed PTAS, which worst-case time complexity can be expressed as O(κ!κkn log n), where κ is the number of the long critical jobs (κ << ν) and the corresponding approximation factor is 1 + 1/k, where κ < k. This is already better than the time complexity of the earlier known approximation schemes. However, the employed algorithmic framework allows a deeper analysis of the running time of PTAS. In particular, we show that the probability that a considerable number of permutations (far less than κ!) are enumerated is close to 0. Hence, with a high probability, the running time of PTAS is fully polynomial.