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 for integer due dates to the scheduling problem, where along with precedence constraints given on the set 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 . We prove that the extended scheduling problem with integer release times of the jobs to minimize schedule length may be solved as an optimal mixed graph coloring problem that consists of the assignment of a minimal number of colors (positive integers) to the vertices of the mixed graph such that, if two vertices and are joined by the edge , their colors have to be different. Further, if two vertices and are joined by the arc , the color of vertex has to be no greater than the color of vertex . We prove two theorems, which imply that most analytical results proved so far for optimal colorings of the mixed graphs , have analogous results, which are valid for the extended scheduling problems to minimize the schedule length or maximal lateness, and vice versa.

Special Issue “2021 Selected Papers from Algorithms’ Editorial Board Members”

“…The paper [7] deals with the problem of scheduling multiprocessor tasks with given precedence constraints and equal processing times in order to minimize the maximum lateness L max . In particular, it is shown that any optimal coloring of a mixed graph is equivalent to the problem of finding an optimal schedule for partially ordered multiprocessor tasks with unit processing times and the minimization of L max .…”