A new approximation algorithm for UET-scheduling with chain-type precedence constraints

Jansen et al. [Jansen, K., Woeginger, G. J. and Yu, Z., UET-Scheduling with chain-type precedence constraints. Report 277, Technische Universität Graz, Institut für Mathematik, Graz, Austria, 1993.] studied the U2Chain problem in which two processors are given and all jobs have unit processing time; every job can be executed on only one machine; there are chain-type precedence constraints among the jobs and the objective is to minimize the makespan. This problem arises from the job-system where different jobs require different resources, thus a job may be processed only on some special machine. Since the problem is strongly NP-hard, they gave an approximation algorithm with performance ratio of 5/3. The purpose of our paper is to improve their result by giving a new approximation algorithm with a worst-case guarantee 3/2 for this problem. In this paper a new approximation algorithm with worst case performance ratio 3/2 is presented for the problem of scheduling unit execution time jobs on two parallel machines with the objective of minimizing the makespan where each job can be processed on only one machine and there are chain-type precedence constraints in the jobs. This result answers an open problem in Jansen et al. [Jansen, K., Woeginger, G. J. and Yu, Z., UET-Scheduling with chain-type precedence constraints. Report 277, Technische Universität Graz, Institut für Mathematik, Graz, Austria, 1993.]: “Does there exist a polynomial time heuristic with worst-case guarantee better than 5/3 for the problem U2Chain?”. The new approximation algorithm is also applied to the problem J2pmtnCmax.