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 9 5 -approximation algorithm for the problem with a fixed sequence. Then we show that the tight approximation ratios of the algorithm are 7 4 and 5 3 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 5 3 -approximation algorithm. • Consider the problem of scheduling with conflict constraints on two parallel dedicated machines. • Present a 9 5 -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 5 3 -approximation algorithm. [ABSTRACT FROM AUTHOR]