Abstract
In this paper, we consider parallel-machine scheduling with release times and submodular penalties (P|rj,reject|Cmax+π(R)), in which each job can be accepted and processed on one of m identical parallel machines or rejected, but a penalty must paid if a job is rejected. Each job has a release time and a processing time, and the job can not be processed before its release time. The objective of P|rj,reject|Cmax+π(R) is to minimize the makespan of the accepted jobs plus the penalty of the rejected jobs, where the penalty is determined by a submodular function. This problem generalizes a multiprocessor scheduling problem with rejection, the parallel-machine scheduling with submodular penalties, and the single machine scheduling problem with release dates and submodular rejection penalties. In this paper, inspired by the primal-dual method, we present a combinatorial 2-approximation algorithm to P|rj,reject|Cmax+π(R). This ratio coincides with the best known ratio for the parallel-machine scheduling with submodular penalties and the single machine scheduling problem with release dates and submodular rejection penalties.
Highlights
In this paper, we consider parallel-machine scheduling with release times and submodular penalties (P|r j, reject|Cmax + π ( R)), in which each job can be accepted and processed on one of m identical parallel machines or rejected, but a penalty must paid if a job is rejected
They proved that this problem is NP-hard, and presented a 2-approximation algorithm and a fully polynomial-time approximation scheme (FPTAS)
We investigate parallel-machine scheduling with release times and submodular penalties (P|r j, reject|Cmax + π ( R)), which is a generalization of parallel-machine scheduling with release times and rejection penalties and single machine scheduling with release dates and submodular penalties
Summary
All jobs must be accepted and processed in classical scheduling problems [1,2,3,4]. to gain more profit, we can reject some jobs that have a larger processing time and result in smaller profits. Zhong et al [10] considered two parallel-machine scheduling with release dates and rejection, and presented a (3/2+ε)-approximation algorithm with time complexity O(( nε )2 ), where ε is any given small positive constant. Zhang et al [15] considered precedence-constrained scheduling with submodular rejection on parallel machines, and proposed a 3-approximation algorithms. Based on the primal-dual method, Liu and Li presented a 2-approximation algorithm for [16] single machine scheduling with release dates and submodular rejection penalty. In this paper, we present a combinatorial 2-approximation algorithm for P|r j , reject| Cmax + π ( R) This ratio coincides with the best known ratio for the parallel-machine scheduling with submodular penalties and the single machine scheduling problem with release dates and submodular rejection penalties.
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