In this paper, we study a vector scheduling problem with rejection on a single machine, in which each job is characterized by a d-dimension vector and a penalty, in the sense that, jobs can be either rejected by paying a certain penalty or assigned to the machine. The objective is to minimize the sum of the maximum load over all dimensions of the total vector of all accepted jobs, and the total penalty of rejected jobs. We prove that the problem is NP-hard and design two approximation algorithms running in polynomial time. When d is a fixed constant, we present a fully polynomial time approximation scheme.
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We are grateful to the anonymous referees for numerous helpful comments and suggestions which helped to improve the presentation of our work. The work is supported in part by the National Natural Science Foundation of China [Nos. 61662088, 11301466], the Natural Science Foundation of Yunnan Province of China [No. 2014FB114], IRTSTYN, and Program for Excellent Young Talents, Yunnan University.
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