Approximation algorithms for stochastic scheduling problems

• Authors: Brian Christopher Dean;Michel X. Goemans
• Venue: Approximation algorithms for stochastic scheduling problems
• Year: 2005

In this dissertation we study a broad class of stochastic scheduling problems characterized by the presence of hard deadline constraints. The input to such a problem is a set of jobs, each with an associated value, processing time, and deadline. We would like to schedule these jobs on a set of machines over time. In our stochastic setting, the processing time of each job is random, known in advance only as a probability distribution (and we make no assumptions about the structure of this distribution). Only after a job completes do we know its actual “instantiated” processing time with certainty. Each machine can process only a singe job at a time, and each job must be assigned to only one machine for processing. After a job starts processing we require that it must be allowed to complete—it cannot be canceled or “preempted” (put on hold and resumed later). Our goal is to devise a scheduling policy that maximizes the expected value of jobs that are scheduled by their deadlines. A scheduling policy observes the state of our machines over time, and any time a machine becomes available for use, it selects a new job to execute on that machine. Scheduling policies can be classified as adaptive or non-adaptive based on whether or not they utilize information learned from the instantiation of processing times of previously-completed jobs in their future scheduling decisions. A novel aspect of our work lies in studying the benefit one can obtain through adaptivity, as we show that for all of our stochastic scheduling problems, adaptivity can only allow us to improve the expected value obtained by an optimal policy by at most a small constant factor. All of the problems we consider are at least NP-hard since they contain the deterministic 0/1 knapsack problem as a special case. We therefore seek to develop approximation algorithms: algorithms that run in polynomial time and compute a policy whose expected value is provably close to that of an optimal adaptive policy. For all the problems we consider, we can approximate the expected value obtained by an optimal adaptive policy to within a small constant factor (which depends on the problem under consideration, but is always less than 10). A small handful of our results are pseudo-approximation algorithms, delivering an approximately optimal policy that is feasible with respect to a slightly expanded set of deadlines. (Copies available exclusively from MIT Libraries, Rm. 14-0551, Cambridge, MA 02139-4307. Ph. 617-253-5668; Fax [phone].) (Abstract shortened by UMI.)