Scheduling to minimize power consumption using submodular functions

Abstract

We develop logarithmic approximation algorithms for extremely general formulations of multiprocessor multi-interval offline task scheduling to minimize power usage. Here each processor has an arbitrary specified power consumption to be turned on for each possible time interval, and each job has a specified list of time interval/processor pairs during which it could be scheduled. (A processor need not be in use for an entire interval it is turned on.) If there is a feasible schedule, our algorithm finds a feasible schedule with total power usage within an O(log n) factor of optimal, where n is the number of jobs. (Even in a simple setting with one processor, the problem is Set-Cover hard.) If not all jobs can be scheduled and each job has a specified value, then our algorithm finds a schedule of value at least (1 - c)Z and power usage within an O(log(1/E)) factor of the optimal schedule of value at least Z, for any specified Z and c > 0. At the foundation of our work is a general framework for logarithmic approximation to maximizing any submodular function subject to budget constraints. We also introduce the online version of this scheduling problem, and show its relation to the classical secretary problem. In order to obtain constant competitive algorithms for this online version, we study the secretary problem with submodular utility function. We present several constant competitive algorithms for the secretary problem with different kinds of utility functions.