ELSEVIER Operations Research Letters 20 (1997) 75-79 A polynomial approximation scheme for problem F2/ /Cmax Mikhail Y. Kovalyov a,*, Frank Werner b alnstitute of Engineering Cybernetics, Belarus Academy of Sciences, Suganova 6, 220012 Minsk, Belarus bOtto-von-Guericke-Universitiit Magdeburg, Fakultgitfiir Mathematik, PSF 4120, Magdeburg, Germany Received 1 January 1995; revised 1 August 1996 Abstract We present a polynomial approximation scheme {H,} for the strongly NP-hard problem of scheduling n jobs in a two- machine flow-shop subject to release dates. This scheme is based on a dynamic programming approach applied to a problem with rounded release dates and processing times. In comparison with Hall's polynomial approximation scheme, our scheme has a better time complexity estimation for small e and sufficiently large n. Keywords: Scheduling; Two-machine flow-shop; Polynomial approximation scheme; Dynamic programming I. Introduction We consider the following flow-shop scheduling problem. There are n independent non-preemptive jobs to be scheduled for processing on machines A and B. Each machine can handle at most one job at a time. Job j must be processed aj >>, 0 time units on machine A and then bj >10 time units on machine B. The processing of job j cannot be started before a re- lease date ~. ~> 0 and the processing of a job does not necessarily have to start on machine B immediately after its completion on machine A. The objective is to minimize the length of the schedule, i.e. the time at which the last job on machine B is completed. We denote the length of an arbitrary schedule S by C(S) and the length of an optimal schedule by C*. According to the traditional notation for scheduling problems of Graham et al. [2], this problem is denoted by F2/rj/Cmax. The problem is strongly NP-hard (see * Corresponding author. [10]), therefore there is no pseudopolynomial algo- rithm or fully polynomial approximation scheme for this problem unless P = NP [1]. In this paper, we develop a polynomial approxima- tion scheme for the problem F2/r//Cmax, i.e. a fam- ily {He} of algorithms with the property that, for any > 0, H~ produces a schedule of a length at most (1 + e)C* and it has a running time polynomial in n and exponential in 1/e. As far as we know, there are only the following approximation results for the problem F2/rj/Cmax. Potts [ 11] investigated the performance of five heuris- tics and showed that the best one guarantees a relative error e = 2. Hall [3] constructed a polynomial approximation scheme with O((1 + 2/e) ~12+6e)/~2(1 + (12+6e)/fz)l+2/~nlogn) running time of each algorithm. This result is based on the notion of a (1 + e)-approximation outline scheme introduced by Hall and Shmoys [4-6] for solving single-machine scheduling problems. An outline scheme is a label- ing of feasible solutions such that, for each feasible 0167-6377/97/$17.00 Copyright (~ 1997 Elsevier Science B.V. All rights reserved PII S01 67-63 77( 96 )00049-!