A Makespan-Optimal Schedule for Processing Jobs with Possible Operation Preemptions As an Optimal Mixed Graph Coloring

Abstract

This paper establishes a relationship between optimal scheduling problems with the minimum schedule length and the problems of finding optimal (strict) colorings of mixed graph vertices, i.e., assigning a minimal set of ordered colors to the vertices V = {${{{v}}_{1}}$, …, ${{{v}}_{{\left| V \right|}}}$} of a mixed graph G = (V, A, E) so that the vertices ${{{v}}_{i}}$ and ${{{v}}_{j}}$ incident to an edge [${{{v}}_{i}}$, ${{{v}}_{j}}$] ∈ E will have different colors and the color of the vertex ${{{v}}_{k}}$ in an arc (${{{v}}_{k}}$, ${{{v}}_{l}}$) ∈ A will be not greater (smaller) than that of the vertex ${{{v}}_{l}}$. As shown below, any optimal coloring problem for the vertices of a mixed graph G can be represented as the problem G_cMPT |[p_ij], pmtn|C_max of constructing a makespan-optimal schedule for processing a partially ordered set of jobs with integer durations p_ij of their operations with possible preemptions. In contrast to classical scheduling problems, executing an operation in the problem G_cMPT |[p_ij], pmtn|C_max may require several machines and, besides the two types of precedence relations defined on the set of operations, unit-time operations of a given subset must be executed simultaneously. The problem G_cMPT |[p_ij], pmtn|C_max is pseudopolynomially reduced to the problem of finding an optimal coloring of the vertices of a mixed graph G (the input data of the scheduling problem). Due to the assertions proved, the results obtained for the problem G_cMPT |[p_ij], pmtn|C_max have analogs for the corresponding optimal coloring problems for the vertices of a mixed graph G, and vice versa.