Complexity and approximation algorithms for two parallel dedicated machine scheduling with conflict constraints - Zhang, An; Zhang, Liang; Chen, Yong; Chen, Guangting; Wang, Xing (Theoretical computer science, 2023)
We investigate two parallel dedicated machine scheduling with conflict constraints. The problem of minimizing the makespan has been shown to be NP-hard in the strong sense under the assumption that the processing sequence of jobs on one machine is given and fixed a priori. The problem without any fixed sequence was previously recognized as weakly NP-hard. In this paper, we first present a 95-approximation algorithm for the problem with a fixed sequence. Then we show that the tight approximation ratios of the algorithm are 74 and 53 for two subproblems which remain strongly NP-hard. We also send an improved algorithm with approximation ratio 3−2≈1.586 for one subproblem. Finally, we prove that the problem without any fixed sequence is actually strongly NP-hard, and design a 53-approximation algorithm. •Consider the problem of scheduling with conflict constraints on two parallel dedicated machines.•Present a 95-approximation algorithm for the strongly NP-hard case where jobs on one machine have a fixed processing sequence, and give improvements to its hard subproblems.•Prove that the problem without any fixed sequence is strongly NP-hard too and propose a 53-approximation algorithm.