An Asymptotically Optimal On-Line Algorithm for Parallel Machine Scheduling

Jobs arriving over time must be non-preemptively processed on one of m parallel machines, each of which running at its own speed, so as to minimize a weighted sum of the job completion times. In this on-line environment, the processing requirement and weight of a job are not known before the job arrives. The Weighted Shortest Processing Requirement (WSPR) on-line heuristic is a simple extension of the well known WSPT heuristic, which is optimal for the single machine problem without release dates. We prove that the WSPR heuristic is asymptotically optimal for all instances with bounded job processing requirements and weights. This implies that the WSPR algorithm generates a solution whose relative error approaches zero as the number of jobs increases. Our proof does not require any probabilistic assumption on the job parameters and relies extensively on properties of optimal solutions to a single machine relaxation of the problem.Singapore-MIT Alliance (SMA

A PTAS for Minimizing Average Weighted Completion Time With Release Dates on Uniformly Related Machines

A classical scheduling problem is to find schedules that minimize average weighted completion time of jobs with release dates. When multiple machines are available, the machine environments may range from identical machines (the processing time required by a job is invariant across the machines) at one end, to unrelated machines (the processing time required by a job on any machine is an arbitrary function of the specific machine) at the other end of the spectrum. While the problem is strongly NP-hard even in the case of a single machine, constant factor approximation algorithms have been known for even the most general machine environment of unrelated machines. Recently, a polynomial-time approximation scheme (PTAS) was discovered for the case of identical parallel machines [1]. In contrast, it is known that this problem is MAX SNP-hard for unrelated machines [10]. An important open problem is to determine the approximability of the intermediate case of uniformly related machines where each machine i has a speed si and it takes p/si time to executing a job of processing size pIn this paper, we resolve this problem by obtaining a PTAS for the problem. This improves the earlier known ratio of (2 + ∈) for the problem

Minimización del tiempo total de flujo de tareas en una sola máquina: Estado del arte

La programación de operaciones en una sola máquina es un problema clásico de la investigación de operaciones. Numerosos métodos han sido propuestos para resolver diferentes instancias del problema, dependiendo de las restricciones impuestas y del objetivo del mismo. En este artículo estamos interesados en ilustrar el estado actual de desarrollo de los métodos y algoritmos existentes en la literatura para el problema de minimización del flujo total de tareas sujetas a fechas de llegadas, tanto en problemas estáticos como dinámicos. Además, las posibilidades de trabajo y las preguntas abiertas serán igualmente expuestas

Improved Bounds on Relaxations of a Parallel Machine Scheduling Problem

We consider the problem of scheduling n jobs with release dates on m identical parallel machines to minimize the average completion time of the jobs. We prove that the ratio of the average completion time of the optimal nonpreemptive schedule to that of the optimal preemptive schedule is at most 7}{3}, improving a bound of (3- 1}{m}) due to Phillips, Stein and Wein. We then use our technique to give an improved bound on the quality of a linear programming relaxation of the problem considered by Hall, Schulz, Shmoys and Wein

Single Machine Scheduling with Release Dates

We consider the scheduling problem of minimizing the average weighted completion time of n jobs with release dates on a single machine. We first study two linear programming relaxations of the problem, one based on a time-indexed formulation, the other on a completiontime formulation. We show their equivalence by proving that a O(n log n) greedy algorithm leads to optimal solutions to both relaxations. The proof relies on the notion of mean busy times of jobs, a concept which enhances our understanding of these LP relaxations. Based on the greedy solution, we describe two simple randomized approximation algorithms, which are guaranteed to deliver feasible schedules with expected objective value within factors of 1.7451 and 1.6853, respectively, of the optimum. They are based on the concept of common and independent a-points, respectively. The analysis implies in particular that the worst-case relative error of the LP relaxations is at most 1.6853, and we provide instances showing that it is at least e/(e - 1) 1.5819. Both algorithms may be derandomized, their deterministic versions running in O(n2 ) time. The randomized algorithms also apply to the on-line setting, in which jobs arrive dynamically over time and one must decide which job to process without knowledge of jobs that will be released afterwards

The Power of α-Points in Preemptive Single Machine Scheduling

We consider the NP-hard preemptive single machine scheduling problem to minimize the total weighted completion time subject to release dates. A natural extension of Smith's ratio rule is to preempt the currently active job whenever a new job arrives that has higher ratio of weight to processing time. We prove that the competitive ratio of this simple on-line algorithm is precisely~2. We also show that list scheduling in order of random α-points drawn from the same schedule results in an on-line algorithm with competitive ratio~4/3. Since its analysis relies on a well-known integer programming relaxation of the scheduling problem, the relaxation has performance guarantee~4/3 as well. On the other hand, we show that it is at best an~8/7-relaxation