A Combinatorial 2-Approximation Algorithm for the Parallel-Machine Scheduling with Release Times and Submodular Penalties

Wencheng Wang,Xiaofei Liu

doi:10.3390/math10010061

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

Introduction

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.

Terminologies and Key Lemmas

Approximation Algorithm

Conclusions