Complexity of the exact domatic number problem and of the exact conveyor flow shop problem

We prove that the exact versions of the domatic number problem are complete for the levels of the boolean hierarchy over NP. The domatic number problem, which arises in the area of computer networks, is the problem of partitioning a given graph into a maximum number of disjoint dominating sets. This number is called the domatic number of the graph. We prove that the problem of determining whether or not the domatic number of a given graph is exactly one of ¢ given values is complete for BH£¥¤§ ¦ NP ¨ , the ©� ¢ th level of the boolean hierarchy over NP. In particular, for ¢��� � , it is DP-complete to determine whether or not the domatic number of a given graph equals exactly a given integer. Note that DP � BH£� ¦ NP ¨. We obtain similar results for the exact versions of generalized dominating set problems and of the conveyor flow shop problem, which arises in real-world applications in the wholesale business, where warehouses are supplied with goods from a central storehouse. Our reductions apply Wagner’s conditions sufficient to prove hardness for the levels of the boolean hierarchy over NP