Lower Bounds & Competitive Algorithms for Online Scheduling of Unit-Size Tasks to Related Machines

In this paper we study the problem of assigning unit-size tasks to related machines when only limited online information is provided to each task. This is a general framework whose special cases are the classical multiple-choice games for the assignment of unit-size tasks to identical machines. The latter case was the subject of intensive research for the last decade. The problem is intriguing in the sense that the natural extensions of the greedy oblivious schedulers, which are known to achieve near-optimal performance in the case of identical machines, are proved to perform quite poorly in the case of the related machines. In this work we present a rather surprising lower bound stating that any oblivious scheduler that assigns an arbitrary number of tasks to $n$ related machines would need $\Omega\left(\frac{\log n}{\log\!\log n}\right)$ polls of machine loads per task, in order to achieve a constant competitive ratio versus the optimum offline assignment of the same input sequence to these machines. On the other hand, we prove that the missing information for an oblivious scheduler to perform almost optimally, is the amount of tasks to be inserted into the system. In particular, we provide an oblivious scheduler that only uses $\cal{O}(\log\!\log n)$ polls, along with the additional information of the size of the input sequence, in order to achieve a constant competitive ratio vs. the optimum offline assignment. The philosophy of this scheduler is based on an interesting exploitation of the {\sc slowfit} concept ([AAFPW97,BFN00,BCK97]; for a survey see [BY98,Azar98,Sgall98]) for the assignment of the tasks to the related machines despite the restrictions on the provided online information, in combination with a layered induction argument for bounding the tails of the number of tasks passing from slower to faster machines. We finally use this oblivious scheduler as the core of an adaptive scheduler that does not demand the knowledge of the input sequence and yet achieves almost the same performance