Scheduling and admission control

We present algorithms and hardness results for three resource allocation problems. The first is an abstract admission control problem where the system receives a series of requests and wants to satisfy as many as possible, but has bounded resources. Algorithms can have performance guarantees for this problem with respect to either acceptances or rejections. These types of guarantees are incomparable and algorithms having different types of guarantee can have nearly opposite behavior. We give two procedures for combining one algorithm of each type into a single algorithm having both types of guarantee simultaneously. The second problem we consider is scheduling with rejections, a combination of scheduling and admission control. For this problem, each job comes with a rejection cost in addition to the standard scheduling parameters. The system schedules a subset of the jobs and schedule quality is total flow plus the cost of rejected jobs. We give lower bounds on the competitive ratio that can be achieved by any deterministic algorithm. We also give an optimal offline algorithm for unit-length jobs with arbitrary rejection costs and two 2-competitive online algorithms for unit-length jobs. Finally, we show that the offline problem is NP-hard even when each job's rejection cost is proportional to its processing time. Our third problem is power-aware scheduling, where the processor can run at different speeds, with its energy consumption depending on the speeds selected. If schedule quality is measured with makespan, we give linear-time algorithms to compute all non-dominated solutions for the general uniprocessor problem and for the multiprocessor problem when every job requires the same amount of work. We also show that the multiprocessor problem becomes NP-hard when jobs can require different amounts of work. If schedule quality is measured with total flow time, we show that finding the optimal schedule for a particular energy budget is impossible. We do, however, give an arbitrarily-good approximation for scheduling equal-work jobs on a multiprocessor.