On the parameterized tractability of single machine scheduling with rejection

In this paper we study a single machine scheduling problem with rejection. In such a scheduling problem, the scheduler can reject to process a job in the shop at a certain cost. Our objective is to minimize the total completion time of the jobs to be processed in the shop, given that the total rejection cost will not exceed a certain predefined threshold. The problem is known to be NP-hard, and to better investigate its hardness our objective is to study whether the problem becomes tractable when some specific natural parameters are of a limited size. The analysis is done by using the theory of parameterized complexity and includes the following parameters: the cardinality of the set of accepted jobs, the number of different processing times, the number of different rejection costs, the maximal processing time, and the maximal rejection cost. We show that the problem is W[1]-hard for the first parameter, while it is fixed-parameterized tractable for all other parameters.