NP-hardness of shop-scheduling problems with three jobs

Yu.N. Sotskov, N.V. Shakhlevich
1995 Discrete Applied Mathematics  
This paper deals with the problem of scheduling n jobs on m machines in order to minimize the maximum completion time or mean flow time of jobs. We extend the results obtained in Sotskov (1989, 1990, 1991) on the complexity of shop-scheduling problems with n = 3. The main result of this paper is an NP-hardness proof for scheduling 3 jobs on 3 machines, whether preemptions of operations are allowed or forbidden. Our terminology follows the classification of scheduling problems used in [lo, 111.
more » ... hen I'= (1 , . . . , m) for all jobs Ji E J, i.e., the routes are identical, we have a flow-shop *Supported by Deutsche Forschungsgemeinschaft, Project ScheMA, and by the Belarusian Fundamental Research Found, Project F60-242. * Corresponding author. 0166-218X/95/$09.50 0 1995-Elsevier Science B.V. All rights reserved SSDI 0166-218X(93)E0169-Y 238 Yu.N. Sorskov, N.V. Shakhlevich / Discrete Applied Mathematics 59 (1995) 237-266 problem, indicated by n 1 m 1 F I@. When ni and I' may vary per job we have a job-shop problem n 1 m I J I @. When the order of the machines in I' is not fixed for any job Ji E J we have an open-shop problem n 1 m ( 0 I @. The parameter @ denotes an optimality criterion of a schedule. If @ = C"" the problem is to find a schedule s* = s*(t) of n jobs minimizing the maximum (total) completion time: Cmax(S*) = maX{fi(S*)IJiEJ}. If Cp = C Ci, the problem is to find a schedule s* = s*(t) of n jobs minimizing the mean flow time: We shall indicate the preemption allowance by a parameter Pr. For example, nlmlJ, PrIGax. The condition tiq > 0 indicates that processing times are strictly postive. There are many efficient algorithms and complexity results for the different cases of scheduling problems under the usual assumption n > m (see [S, 8-11,191). The purpose of this paper is to improve the results obtained in [14-161 on the study of the complexity of shop-scheduling problems with fixed number of jobs when n < m. In Section 2 we prove that the problems 3 13 1 J) C"" 3 ) 3 I J, Pr I C"" 3 13 I J 11 Ci and 3 13 IJ, Pr lCCi are NP-hard. In Section 3 the same results are obtained for jImIF, PrIGax, 3lmlF, PrlCCi, 3 I m I F, Pr, 4, > 0 I Cm",
doi:10.1016/0166-218x(95)80004-n fatcat:sncphigzhbaqhozbdfajmfdf3m