The complexity of shop-scheduling problems with two or three jobs

Abstract

This paper deals with the n/ m/J/ C max problem of optimal makespan scheduling n jobs with fixed routines on m machines. It is shown that the 3/ m/J/ C max problem with identical routines of jobs and the 3/5/J/ C max problem are NP-hard. The same results are obtained for the 3/ m/J/∑ C i problem of minimizing mean flow time of three jobs on m machines. The problem of optimal scheduling of two jobs with any regular criterion is shown to be solved by an O( r 3) algorithm if preemptions of operations are allowed and by an O( r 2 log 2 r) algorithm, otherwise. Here the parameter r denotes the maximal number of operations per job.