Complexity and approximation algorithms for two parallel dedicated machine scheduling with conflict constraints

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.