Machine sequencing: Disjunctive graphs and degree-constrained subgraphs

In two earlier papers, we proposed algorithms for finding an optimal sequence of processing m items on q machines, by finding a minimaximal path in a disjunctive network. In a third paper, this latter model was generalized (from 2-state to 3-state disjunctive graphs) so as to accommodate project scheduling with resource constraints. In this paper, we discuss another algorithm for the (2-state) disjunctive network problem, closely related to those mentioned above. To make the paper self-contained, section 2 briefly describes the problem. Section 3 introduces a class of constraints which forms the basis of the algorithm discussed in section 4. The constraints have only 1, −1, or 0 as coefficients on the left-hand side, integers on the right-hand side. The whole procedure of generating these constraints and finding a feasible solution whenever a new constraint is added, can be interpreted (section 5) as a process of generating a graph with degree-constraints on its nodes, and then finding a subgraph satisfying the degree-constraints. The nodes of the graph are generated by solving a critical-path-problem, the feasible subgraphs are found by implicit enumeration.