Penalty cost constrained identical parallel machine scheduling problem

We consider a version of parallel machine scheduling with rejection. An instance of the problem is given by m identical parallel machines and a set of n   independent jobs, with each job having a processing time and a penalty. A job may be accepted to be processed or be rejected at its penalty. The objective of the problem is to partition the set of jobs into two subsets, the subset of accepted and the subset of rejected jobs, and to schedule the set of accepted jobs such that the makespan is minimized under the constraint that the total penalty of the rejected jobs is no more than a given bound. In this paper, we present a 2-approximation algorithm within strongly polynomial time for the problem. We also present a polynomial time approximation scheme whose running time is O(nmO(1ϵ2)+mn2) for the problem. Moreover, for the case where the number of machines is a fixed constant m  , our results lead to a fully polynomial time approximation scheme for the problem. Our result is fairly good in the sense that in a reasonable size of jobs, our FPTAS improves previous best running time from O(nm+2/ϵm)O(nm+2/ϵm) to O(1/ϵ2m+3+mn2)O(1/ϵ2m+3+mn2).