Devices]: Modes of Computation---Online computation General Terms: ALGORITHMS, THEORY Additional Key Words and Phrases: Stochastic scheduling, Approximation, Worst-case performance, Priority policy, LP-relaxation, WSEPT rule, Asymptotic optimality This research was partially supported by the German-Israeli Foundation for Scientific Research and Development (G.I.F.) under grant I 246-304.02/97. An extended abstract appeared in the Proceedings of the 2nd Int. Workshop on Approximation Algorithms for Combinatorial Optimization Problems (APPROX'99). Authors' addresses: Rolf H. Mohring and Marc Uetz. Technische Universitat Berlin, Fachbereich Mathematik, Sekr. MA 6--1, Straße des 17. Juni 136, 10623 Berlin, Germany, Email: fmoehring, uetzg@math.tu--berlin.de. Andreas S. Schulz. MIT, Sloan School of Management and Operations Research Center, E53--361, 30 Wadsworth St, Cambridge, MA 02139, Email: schulz@mit.edu. Permission to make digital or hard copies of part or all of this work for person...

We study the question of which optimization problems can be optimally or approximately solved by "greedy-like " algorithms. For definiteness, we will limit the present discussion to some well-studied scheduling problems although the underlying issues apply in a much more general setting. Of course, the main benefit of greedy algorithms lies in both their conceptual simplicity and their computational efficiency. Based on the experience from online competitive analysis, it seems plausible that we should be able to derive approximation bounds for "greedy-like " algorithms exploiting only the conceptual simplicity of these algorithms. To this end, we need (and will provide) a precise definition of what we mean by greedy and greedy-like.

We consider the problem of on-line scheduling of jobs arriving one by one on uniformly related machines, with or without preemption. We prove a lower bound of 2, both with and without preemption, for randomized algorithms working for an arbitrary number of machines. For a constant number of machines we give new lower bounds for the preemptive case.

We study the problem of online scheduling on two uniform machines withspeeds 1 and s * 1. A OE ss 1.61803 competitive deterministic algorithm was al-ready known. We present the first randomized results for this problem: We show that randomization does not help for speeds s * 2, but does help for all s! 2. We present a simple memoryless randomized algorithm with competitive ratio (4 \Gamma s)(1 + s)=4 ^ 1.56250. We analyze other randomized algorithms that demonstrate that the randomized competitive ratio is at most 1.52778 for any s. Thesealgorithms are barely random, i.e., they use only a constant number of random bits. Finally, we present a 1 + s/(s² + s + 1) competitive deterministic algorithm for the preemptive version of this problem. For any s, it is best possible evenamong randomized preemptive algorithms.

Abstract. We introduce the online scheduling problem for sorting buffers. A service station and a sorting buffer are given. An input sequence of items which are only characterized by a specific attribute has to be processed by the service station which benefits from consecutive items with the same attribute value. The sorting buffer which is a random access buffer with storage capacity for k items can be used to rearrange the input sequence. The goal is to minimize the cost of the service station, i.e., the number of maximal subsequences in its sequence of items containing only items with the same attribute value. This problem is motivated by many applications in computer science and economics. The strategies are evaluated in a competitive analysis in which the cost of the online strategy is compared with the cost of an optimal offline strategy. Our main result is a deterministic strategy that achieves a competitive ratio of O(log 2 k). In addition, we show that several standard strategies are unsuitable for this problem, i.e., we prove a lower bound of Ω ( √ k) on the competitive ratio of the First In First Out (FIFO) and Least Recently Used (LRU) strategy and of Ω(k) on the competitive ratio of the Largest Color First (LCF) strategy. 1

We study a preemptive semi-online scheduling problem. Jobs with non-increasing sizes arrive one by one to be scheduled on two uniformly related machines. We analyze the algorithms as a function of the speed ratio (q 1) between the two machines. We design algorithms of optimal competitive ratio for all values of q, and show that for q > 2, idle time needs to be introduced. This is the rst preemptive scheduling problem over list, where idle time is provably required.

Dynamism of trade activity inevitably results in situations where sellers face local supply shortages. In such cases, sellers need to decide which buyer purchase requests to satisfy. Commonly, sellers satisfy purchase requests based on their arrival order, i.e., First In is First Served (FIFS). In electronic trade, sellers may follow strategies different from FIFS without the buyers being able to detect this difference.

We study the problem of scheduling a set of preemptive tasks, where each task j is specified by its release time r j , deadline d j , processing time p j , and weight w j representing its profit rate.

The focus of this paper is on how to design evolutionary algorithms (EAs) for solving stochastic dynamic optimization problems online, i.e. as time goes by. For a proper design, the EA must not only be capable of tracking shifting optima, it must also take into account the future consequences of the evolved decisions or actions. A previous framework describes how to build such EAs in the case of non-stochastic problems. Most real-world problems however are stochastic. In this paper we show how this framework can be extended to properly tackle stochasticity. We point out how this naturally leads to evolving strategies rather than explicit decisions. We formalize our approach in a new framework. The new framework and the various sources of problem–difficulty at hand are illustrated with a running example. We also apply our framework to inventory management problems, an important real–world application area in logistics. Our results show, as a proof of principle, the feasibility and benefits of our novel approach.

We study preemptive scheduling on uniformly related processors, where jobs are arriving one by one in an on-line fashion. We consider the class of machine sets where the speed ratios are nondecreasing as speed increases. For each set of machines in this class, we design an algorithm of optimal competitive ratio. This generalizes the known result for identical machines, and solves other interesting cases. 1 Introduction We consider on-line scheduling on m uniformly related machines. Jobs arrive on-line, and each job has to be assigned before the next job arrives. This scheduling model is called "scheduling jobs one by one" (see [9]). Preemption is allowed, hence each job may be cut into a few pieces. These pieces are to be assigned to possibly different machines, in non-overlapping time slots. (Non-preemptive algorithms are not allowed to cut the job and have to assign it continuously to one machine.) School of Computer and Media Sciences, The Interdisciplinary Center, P.O.B. 167, 4...