Optimal Online Algorithms for Minimax Resource Scheduling

Abstract

Abstract. We consider a very general online scheduling problem with an objective to minimize the maximum level of resource allocated. We find a simple characterization of an optimal deterministic online algorithm. We develop further results for two more specific problems, single resource scheduling and hierarchical line balancing. We determine how to compute optimal online algorithms for both problems using linear programming and integer programming, respectively. We show that randomized algorithms can outperform deterministic algorithms, but only if the amount of work done is a non-concave function of resource allocation.Keywords: Online algorithms, competitive analysis, worst-case analysis, single-machine scheduling, multiprocessor scheduling, line balancing 1. Introduction. Consider the following problem: work with different deadlines arrives over time and has to be performed using a resource. The quantities of work that arrive as well as their deadlines only become known at the times of arrival. At a given set of time points, the decision maker decides how much resource to allocate and which of the available work to perform at that time. The objective is to minimize the maximum amount of resource allocated at any time during the planning period. This problem is called the online resource minimization problem (ORMP). It occurs in production scheduling settings in which the major cost component is energy consumption (Kleywegt et al. [13]). The decision maker would like to spread the workload out as evenly as possible over time, but faces the dilemma of uncertainty about future work. That is, the decision maker must make a trade-off between allocating too much resource early, and postponing too much work to be completed with work that arrives later.Consider another problem: work with different requirements arrives over time and has to be assigned to a collection of machines with different capabilities. The machines form a linear hierarchy based on their capabilities, i.e., machine j has at least the same capabilities as machine j − 1. The amount of work that arrives as well as the required machine capabilities only become known at the time of arrival. At a given set of decision points, the decision maker decides how to assign the work that has arrived since the previous decision point to the machines. The objective is to minimize the maximum amount of work assigned to any machine. This problem is called the hierarchical line balancing problem (HLBP).In both the ORMP and the HLBP, the quality of an algorithm is evaluated by its competitive ratio, i.e., the worst-case ratio over all possible instances of the value of the solution produced by the algorithm and the value of the optimal solution with perfect information.In this paper, we introduce a simple parameterized deterministic algorithm, called the α-policy, with parameter α and competitive ratio α, provided it produces a feasible solution. We show that with an appropriate choice of parameter α, the α-policy

Peak Shaving through Resource Buffering

“…In the minimax work-scheduling problem [12], the goal is to minimize the maximum amount of work done in any timeslot over a finite time-horizon. Our online problem is related to a previously studied special case in which jobs with deadlines are assigned online.…”

“…Second, we are allowed to do work (request and store energy) in advance. One online algorithm for the jobs-by-deadlines problem is the α-policy [12]: at each timeslot, the amount of work done is α times the maximum per-unit-timeslot amount of work that OPT would have done, when running on the partial input received so far. One of our online algorithms adopts a similar strategy.…”

When to Reap and When to Sow – Lowering Peak Usage with Realistic Batteries

“…The goal in the minimax work-scheduling problem [7] is to minimize the maximum amount of work done in any timeslot over a finite time-horizon. Our online problem is related to a previously studied special case of this in which jobs with deadlines are assigned online.…”

“…While the problems differ in important respects (see [4]), the objectives are similar. Indeed, while the α-policy of [7] performs α times the maximum per-unit-timeslot amount of work that OPT would have done, when running on the partial input received so far, many of our algorithms ensure that the savings at each point is a multiple of the optimal savings so far. The demand curve (combined with battery information) is the problem instance; the request curve is the problem solution.…”