Scheduling Multiprocessor Tasks with Equal Processing Times as a Mixed Graph Coloring Problem

Abstract

This article extends the scheduling problem with dedicated processors, unit-time tasks, and minimizing maximal lateness Lmax for integer due dates to the scheduling problem, where along with precedence constraints given on the set V={v1,v2, …,vn} of the multiprocessor tasks, a subset of tasks must be processed simultaneously. Contrary to a classical shop-scheduling problem, several processors must fulfill a multiprocessor task. Furthermore, two types of the precedence constraints may be given on the task set V. We prove that the extended scheduling problem with integer release times ri≥0 of the jobs V to minimize schedule length Cmax may be solved as an optimal mixed graph coloring problem that consists of the assignment of a minimal number of colors (positive integers) {1,2, …,t} to the vertices {v1,v2, …,vn}=V of the mixed graph G=(V,A, E) such that, if two vertices vp and vq are joined by the edge [vp,vq]∈E, their colors have to be different. Further, if two vertices vi and vj are joined by the arc (vi,vj)∈A, the color of vertex vi has to be no greater than the color of vertex vj. We prove two theorems, which imply that most analytical results proved so far for optimal colorings of the mixed graphs G=(V,A, E), have analogous results, which are valid for the extended scheduling problems to minimize the schedule length or maximal lateness, and vice versa.