Online Scheduling with Bounded Migration

Consider the classical online scheduling problem where jobs that arrive one by one are assigned to identical parallel machines with the objective of minimizing the makespan. We generalize this problem by allowing the current assignment to be changed whenever a new job arrives, subject to the constraint that the total size of moved jobs is bounded by ƒ times the size of the arriving job. Our main result is a linear time ‘online approximation scheme’, that is, a family of online algorithms with competitive ratio FE„W… and constant migration factor ƒ iR…j , for any fixed … †‚k . This result is of particular importance if considered in the context of sensitivity analysis: While a newly arriving job may force a complete change of the entire structure of an optimal schedule, only very limited ‘local’ changes suffice to preserve near-optimal solutions. We believe that this concept will find wide application in its own right. We also present simple deterministic online algorithms with migration factors ƒˆ‡ H and ƒ‰‡ l Š4I , respectively. Their competitive ratio IKŠ4H beats the lower bound on the performance of any online algorithm in the classical setting without migration. We also present improved algorithms and similar results for closely related problems. In particular, there is a short discussion of corresponding results for the objective to maximize the minimum load of a machine. The latter problem has an application for configuring storage servers that was the original motivation for this work. 1 Introduction A classical scheduling problem. One of the most fundamental scheduling problems asks for an assignment of jobs to ‹ identical parallel machines so as to minimize the makespan. (The makespan is the completion time of the last job that finishes in the schedule; it also equals the maximum machine load.) In the standard classification scheme of Graham, Lawler, Lenstra, & Rinnooy Kan [15], this scheduling problem is denoted by Œ:ŒŽ v‘.’ and it is well known to be strongly NP-hard [12]. The offline variant of this problem assumes that all jobs are known in advance whereas in the online variant the jobs are incrementally revealed by an adversary and the online algorithm can only choose the machine for the new job without being allowed to move other jobs. Note that dropping this radical constraint on the online algorithm yields the offline situation. A new online scheduling paradigm. We study a natural generalization of both offline and online problems. Jobs arrive incrementally but, upon arrival of a new job “ , we are allowed to migrate some previous jobs to other machines. The total size of the migrated jobs however must be bounded by ƒ/”C• where ”C• is the size of the new job. For migration factor ƒ–‡ k we get the online setting and for ƒ—‡™˜ we get the offline setting. Approximation algorithms. For an offline optimization problem, an approximation algorithm efficiently (in polynomial time) constructs schedules whose values are within a constant factor š|› F of the optimum solution value. The number š is called performance guarantee or performance ratio of the approximation algorithm. A family of polynomial time approximation algorithms with performance guarantee Fs„€… for all fixed …_†œk is called a polynomial time approximation scheme (PTAS). Competitive analysis. In a similar way, competitive analysis evaluates solutions computed in the online setting. An online algorithm achieves competitive ratio š{› F if it always maintains solutions whose objective values are within a factor š of the offline optimum. Here, in contrast to offline approximation results, the achievable values š are not determined by limited computing power but by the apparent lack of information about parts of the input that will only be revealed in the future. As a consequence, for all interesting classical online problems it is rather easy to come up with lower bounds that create a gap between the best possible competitive ratio š and F . In particular, it is usually impossible to construct a family of i Fo„{…j -competitive online algorithms for such problems. 2 Related Work For the online machine scheduling problem, Graham’s list scheduling algorithm keeps the makespan within a factor H^eFžŠ ‹ of the offline optimum [13]: Schedule a newly arriving job on the least loaded machine. It can also easily be seen that this bound is tight: adversarial sequence consists of ‹ i ‹ ŸFžj jobs of size ¡ followed by one job of size F . The optimal makespan in this case is F . For the offline setting, Graham showed three years later that sorting the jobs in the order of non-increasing size before feeding them to the list scheduling algorithm yields an approximation algorithm with performance ratio l Š4IˆFžŠGiRI ‹ j [14]. Later, exploiting the relationship