An optimal online algorithm for the parallel-batch scheduling with job processing time compatibilities |
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Authors: | Ruyan Fu Ji Tian Shisheng Li Jinjiang Yuan |
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Affiliation: | 1.School of Mathematics,China University of Mining and Technology,Xuzhou,People’s Republic of China;2.Department of Information and Computation Science,Zhongyuan University of Technology,Zhengzhou,People’s Republic of China;3.School of Mathematics and Statistics,Zhengzhou University,Zhengzhou,People’s Republic of China |
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Abstract: | We consider the online (over time) scheduling on a single unbounded parallel-batch machine with job processing time compatibilities to minimize makespan. In the problem, a constant \(\alpha >0\) is given in advance. Each job \(J_{j}\) has a normal processing time \(p_j\). Two jobs \(J_i\) and \(J_j\) are compatible if \(\max \{p_i, p_j\} \le (1+\alpha )\cdot \min \{p_i, p_j\}\). In the problem, mutually compatible jobs can form a batch being processed on the machine. The processing time of a batch is equal to the maximum normal processing time of the jobs in this batch. For this problem, we provide an optimal online algorithm with a competitive ratio of \(1+\beta _\alpha \), where \(\beta _\alpha \) is the positive root of the equation \((1+\alpha )x^{2}+\alpha x=1+\alpha \). |
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