Mixed graph colouring as scheduling multi-processor tasks with equal processing times

A problem of scheduling partially ordered unit-time tasks processed on dedicated machines is formulated as a mixed graph colouring problem, i. e., as an assignment of integers (colours) {1, 2, …, t} to the vertices (tasks) V {ν1, ν2, …, νn}, of the mixed graph G = (V, A, E) such that if vertices vp and vq are joined by an edge [νp, νq] ∈ E their colours have to be different. Further, if two vertices νp and νq are joined by an arc (νi, νj) ∈ A the colour of vertex νi has to be no greater than the colour of vertex νj. We prove that an optimal colouring of a mixed graph G = (V, A, E) is equivalent to the scheduling problem GcMPT|pi = 1|Cmax of finding an optimal schedule for partially ordered multi-processor tasks with unit (equal) processing times. Contrary to classical shop-scheduling problems, several dedicated machines are required to process an individual task in the scheduling problem GcMPT|pi = 1|Cmax. Moreover, along with precedence constraints given on the set V {ν1, ν2, …, νn}, it is required that a subset of tasks must be processed simultaneously. Due to the theorems proved in this article, most analytical results that have been proved for the scheduling problems GcMPT |pi = 1|Cmax so far, have analogous results for optimal colourings of the mixed graphs G = (V, A, E), and vice versa.

Two-Machine Job-Shop Scheduling with Equal Processing Times on Each Machine

In this paper, we consider a two-machine job-shop scheduling problem of minimizing total completion time subject to n jobs with two operations and equal processing times on each machine. This problem occurs e.g., as a single-track railway scheduling problem with three stations and constant travel times between any two adjacent stations. We present a polynomial dynamic programming algorithm of the complexity O ( n 5 ) and a heuristic procedure of the complexity O ( n 3 ) . This settles the complexity status of the problem under consideration which was open before and extends earlier work for the two-station single-track railway scheduling problem. We also present computational results of the comparison of both algorithms. For the 30,000 instances with up to 30 jobs considered, the average relative error of the heuristic is less than 1 % . In our tests, the practical running time of the dynamic programming algorithm was even bounded by O ( n 4 ) .

Online Parallel-Machine Scheduling in KRT Environment to Minimize Total Weighted Completion Time

This paper studies online scheduling on [Formula: see text] identical parallel machines under the KRT environment, where jobs arrive over time and “KRT” means that in the online setting no jobs can be released when all of the machines are busy. The goal is to determine a feasible schedule to minimize the total of weighted completion times. When [Formula: see text], we prove that WSPT is an optimal online algorithm. When [Formula: see text], we first present a lower bound [Formula: see text], and then show that WSPT is a 2-competitive online algorithm for the case [Formula: see text]. For the case in which [Formula: see text] and all jobs have equal processing times, we provide a best possible online algorithm with a competitive ratio of [Formula: see text].